= Solution
In <Fourier transform spectroscopy>, a scanning <Michelson interferometer> measures <intensity> against <optical path difference> $\ell$ rather than selecting one narrow <wavelength> at a time. After subtraction of the constant background, an ideal stationary source gives an <interferogram>
$$
J(\ell)=\int_0^\infty S(\nu)\cos(2\pi\nu\ell/c)\,d\nu.
$$
A cosine <Fourier transform> recovers the spectral <power> density $S(\nu)$ after instrumental calibration. A mirror <displacement> $x$ changes the round-trip path by $2x$; the zero-path-difference burst aligns the scan, and a reference laser can calibrate the <displacement>.
Finite maximum path difference $L$ multiplies the <interferogram> by a window and convolves the recovered <radiation spectrum> with that window's <Fourier transform>. Thus \b[$\delta\nu$ is of order $c/L$]; precise factors depend on scan range and resolution convention. Uniform path samples of separation $\Delta\ell$ need \b[$\nu_{\max}<c/(2\Delta\ell)$] to avoid <aliasing> without bandpass information. <interferogram apodization> reduces sidelobes at the price of broader lines. The broad accepted optical throughput can be advantageous; the multiplex signal-to-noise advantage depends on detector noise and is not universal in photon-noise-limited measurements. Source drift during a scan, finite beam divergence, imperfect sampling and <wave phase> errors require control. The finite-window reasoning is the <delay-range resolution in Fourier transform spectroscopy>.<Introduction to Fourier Transform Spectroscopy by Julius Cohen>[NBSIR 86-3339, chapters 13–16, printed pp.51–80: basic equation, apodization, resolution and sampling]{ref}
Back to article page
