= Spectrometer and resolution
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>.
\Image[/past-exam-of-the-optics-and-electrodynamics-course-of-the-university-of-cambridge/2018-b2-fourier-spectrometer.png]
{title=Scanning Michelson spectrometer: mirror position sets delay, and a Fourier transform of the recorded interferogram recovers the spectrum}
For matched arm polarizations and balanced throughput, write the ideal signal as
$$
I(\tau)=I_{\rm dc}+C\int_0^\infty S(\omega)\cos(\omega\tau)\,d\omega,
$$
where $S(\omega)$ 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 $2L/c$, so
$$
\delta\nu\sim\frac{c}{2L},\qquad
\delta\lambda\simeq\frac{\lambda^2}{c}\delta\nu,
\qquad \boxed{\delta\lambda\propto\frac{\lambda^2}{L}.}
$$
At 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.
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