= Solution
A <Michelson interferometer> divides a source beam at a partially reflecting <beam splitter>. The two perpendicular arms terminate in <mirrors> whose reflected beams return to the splitter and overlap at the detector. The recombined fields must have overlapping <wavefronts> and compatible <polarization of waves>. Moving one <mirror> by $\Delta L$ changes the round-trip <optical path difference> by $2\Delta L$, so $N$ passing fringes determine $\Delta L=N\lambda/2$. Conversely a known <mirror> <displacement> measures <wavelength>. Scanning the delay can determine <temporal optical coherence> and, by <Fourier transform spectroscopy>, spectral structure. A compensating plate may equalize glass traversal, particularly for broadband illumination.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2015/paper-2-michelson.png]
{title=Michelson optical arrangement and the calculated top-hat-spectrum interferogram}
For equal real peak field amplitudes, the superposition is
$$
E(t)=a\cos\omega t+a\cos(\omega t+\phi)
=2a\cos(\phi/2)\cos(\omega t+\phi/2).
$$
Time averaging the square over many optical cycles gives $\langle E^2\rangle=2a^2\cos^2(\phi/2)=a^2(1+\cos\phi)$. With $\phi=kx$ after absorbing a constant port-dependent phase into the zero of $x$,
$$
\boxed{I(x)=I_0[1+\cos(kx)].}
$$
Here $I_0$ is the sum of the two separate-arm <intensities>; maximum <intensity> is $2I_0$ and the ideal minimum is zero. Equal amplitudes give unit <interference visibility>.
For a stationary source with a flat <intensity> spectrum, different spectral components add in time-averaged <intensity>. Keeping their individual two-arm interference terms and defining $\operatorname{sinc}z=\sin z/z$ gives
$$
I(x)=C\int_{k_0-\Delta k/2}^{k_0+\Delta k/2}[1+\cos(k'x)]\,dk'
=C\Delta k\left[1+\cos(k_0x)\operatorname{sinc}\left(\frac{x\Delta k}{2}\right)\right].
$$
One must not coherently add unrelated source <frequencies> as if they had a fixed common phase. The sinc envelope is the <Fourier transform> of the rectangular spectral <intensity>.
Numerically, $\nu_0=5.00\times10^{14}\,\mathrm{Hz}$ and $\Delta\nu=1.00\times10^{13}\,\mathrm{Hz}$. The annotated interferogram uses
$$
\boxed{\frac{I(x)}{C\Delta k}=1+\cos\left(\frac{2\pi\nu_0x}{c}\right)
\operatorname{sinc}\left(\frac{\pi\Delta\nu x}{c}\right).}
$$
It is even in $x$, has central value 2 and background 1, and its carrier-fringe period in <optical path difference> is $c/\nu_0=0.600\,\mathrm{\mu m}$. The first envelope zeros are at \b[$x=\pm c/\Delta\nu=\pm30.0\,\mathrm{\mu m}$], corresponding to <mirror> shifts of $\pm15.0\,\mathrm{\mu m}$ from equal paths. Side lobes persist beyond the zeros; a negative sinc reverses fringe phase. The <intensity> remains between $1-|\operatorname{sinc}|$ and $1+|\operatorname{sinc}|$, not between the signed sinc curves with an assumed fixed ordering.
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