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A non-polarizing beam splitter divides the incident spectrum into a fixed arm and an arm with a translating mirror. Recombining their returned fields gives a detector signal as a function of mirror position $x$. The round-trip optical path difference is $D=2x$ relative to the equal-arm position, so the delay is $\tau =2x/c$ in air approximated as vacuum. Record both detector signal and calibrated position, subtract the constant background, and take a Fourier transform of the interferogram.

For matched arm polarizations and balanced throughput, write the ideal signal aswhere $S(\omega )$ is the incident spectral power density and $C$ is a constant set by the beam splitter and detector. This is the cosine-transform form of Fourier transform spectroscopy, or equivalently the Wiener-Khinchin theorem applied to delayed self-interference.

The delay-range resolution in Fourier transform spectroscopy follows because recording a finite delay range multiplies the correlation by a window. Its transform broadens each spectral line by an instrumental width of order the reciprocal delay range. A mirror travel range $L$ creates a delay range of order $2L/c$, soAt fixed wavelength, absolute wavelength resolution therefore improves inversely with travel range. The numerical coefficient depends on whether $L$ is a one-sided or total travel, on the window or apodization, and on whether resolution means first zero, half width or another criterion. The proportionality is the requested convention-independent result. Position sampling must also be fine enough to sample the fastest fringe variation without aliasing.

For mutually incoherent spectral components, the detected intensities add. If $S(\nu )$ describes the mean output spectral intensity, the interferogram isSubtract the constant background, scan the mirror with a calibrated displacement, and take the Fourier transform with respect to delay $\tau =x/c=2d/c$. This is Fourier transform spectroscopy. The transform recovers the spectrum subject to detector response, finite scan length and any window function. Sampling the displacement finely enough to resolve the shortest optical fringes avoids aliasing.

Using the usual reciprocal-maximum-delay resolution estimate, $\delta \nu \simeq c/x_{\max }$, and $|\delta \nu |\simeq c\delta \lambda /{\lambda }^2$, givesThe mirror displacement is measured from zero optical path difference. This delay-range resolution in Fourier transform spectroscopy is a conventional estimate: a specified window and resolution definition introduce a numerical factor. For example, a symmetric rectangular delay window has its first instrumental zero at $\delta \nu =1/(2{\tau }_{\max })$, which would give $6.25\mathrm{mm}$ under that particular criterion. The paper does not specify a line-profile criterion; the standard $12.5\mathrm{mm}$ estimate states its convention explicitly.