Fourier transform spectroscopy

Fourier transform spectroscopy measures an interferogram as a function of optical path length difference. Its Fourier transform recovers the spectrum. An imaging implementation obtains an interferogram at every detector pixel, but a scanned cube is not simultaneous at all wavelengths.

Multiplying a finite interferogram by a smooth window reduces sharp-edge artifacts. In the recovered radiation spectrum this convolves the true spectral density with the window's Fourier transform. Smoother windows generally lower sidelobes while broadening the main lobe, so a resolution figure must specify the window and criterion. Apodization does not recover unmeasured long-delay information.^[ref]

For $N$ equally intense spectral lines at ${\omega }_n={\omega }_0+n\Delta \omega$, sum their delayed self-correlations as a geometric series. The result isThe interferometric visibility revives every $T=2\pi /\Delta \omega$. There are $N-1$ zeros between adjacent revivals, at $\tau =jT/N$ for $j=1,\dots ,N-1$. The removable singularity at a revival has unit modulus. Revival spacing fixes line spacing; the internal zero count fixes $N$ under the equal-intensity model. The interferogram alone does not establish a fixed relative phase between the source lines or prove the emission is a coherent pulse train.

A Michelson interferometer mirror displacement $x$ changes round-trip optical path difference by $2x$, giving delay $\tau =2x/c$. An interferogram recorded over a finite delay range is multiplied by a window; the recovered spectrum is the true spectrum convolved with that window's Fourier transform. Its characteristic frequency resolution scales inversely with delay range. Consequently a mirror travel range $L$ gives $\delta \nu \propto c/L$ and $\delta \lambda \simeq ({\lambda }^2/c)\delta \nu \propto {\lambda }^2/L$. Numerical width factors depend on the scan definition, window and resolution criterion; the inverse-travel scaling does not.