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An astronomical interferometer combines light from collectors separated by projected interferometric baseline $\mathbf{B}$. For a distant, narrowband, spatially incoherent source, the Van Cittert-Zernike theorem gives normalized complex visibilityThe resolving scale is $\lambda /B$, while the individual collector and beam combiner restrict the field of view. Delay lines compensate geometric path differences. To see broadband fringes, residual optical path length mismatch must lie within the coherence length of order ${\lambda }^2/\Delta \lambda$; fringe tracking follows rapidly changing atmospheric piston in interferometry. Baseline changes and Earth-rotation aperture synthesis sample additional spatial frequencies.

For an ideal uniform circular star of full angular diameter $\Theta$,Real limb darkening changes the conversion. Several baselines constrain binaries, stellar radii and circumstellar structure; orbital angular separations combined with radial velocities can give physical orbital scales. calibration stars test contrast losses, alongside wavelength, photometric imbalance, polarization and bandwidth controls. For three collectors the closure phase cancels telescope-based phase errors: ${\varphi }_{12}+{\varphi }_{23}+{\varphi }_{31}$ retains source asymmetry information. Sparse baseline coverage still leaves image/model ambiguities.

In stellar intensity interferometry, thermal-light correlations instead give $g^{(2)}(0)-1=|g^{(1)}{|}^2$ for an ideal single-polarization resolved temporal mode. The Sirius experiment correlated separate detector outputs rather than optically combining the beams.^[ref] Pure atmospheric phase shifts do not alter local intensities, but finite detector bandwidth dilutes the correlation and photon statistics limit sensitivity; the two-detector observable also loses the complex phase.

Temporal coherence compares the field at different times at one location: a narrow spectrum maintains a predictable phase relation over a longer delay. Spatial coherence compares the field at different locations at the same time: different angular source components illuminate a baseline with different phase differences, which wash out interference when averaged. These are properties of correlations, not claims that all thermal source points have a common phase.

Assume a distant spatially incoherent, quasimonochromatic source, small angles, identical slit transmission, matched polarizations and equal mean intensity at the two slits. For an angular component $\theta$, the relative input phase is $-kd\theta$. Let $\phi$ be the additional observation-plane phase. Incoherent angular components add intensities, givingIts extrema are $2F(0)\pm 2|F(d)|$, henceFor symmetric $I(\theta )$, $F(d)$ is a real cosine transform. Physical visibility is its normalized absolute value; the printed proportionality omits that qualification. A negative signed coherence reverses fringe phase rather than giving negative contrast. This is the small-angle Van Cittert-Zernike theorem. Temporal coherence analogously depends on the Fourier transform of the spectrum.

A Michelson stellar interferometer uses separated collecting mirrors to obtain a large adjustable projected baseline without requiring a monolithic objective of that diameter. Each collector supplies light to a common combiner. Optical path compensation keeps the delay within the temporal coherence range; varying mirror separation samples spatial coherence and resolves stellar angular structure finer than a small collector alone.

Let $g_{\star }(Y)$ be the normalized signed coherence of one star, and take the baseline along the projected binary separation. The equal-brightness binary-star visibility for identical profiles centred at $\pm \beta /2$ giveFor uniform circular discs of angular diameter $\alpha$,For binary separation from visibility-zero spacing, the rapid binary visibility zeros satisfy $Y_j=(j+\frac{1}{2})\lambda /\beta$, so their repeated spacing is $\lambda /\beta$. The slower single-star envelope first vanishes at $Y=1.21967\lambda /\alpha$. The stellar angular diameter from a visibility envelope and the binary separation follow by fitting both scales; multiple baselines distinguish envelope zeros from binary zeros.

Angular width alone does not specify a unique brightness profile. If the intended one-dimensional model is a uniform angular strip of full width $\alpha$, replace the disc factor by $\mathrm{sinc}(\pi \alpha Y/\lambda )$ with $\mathrm{sinc}u=\sin u/u$; its first envelope zero is $\lambda /\alpha$. Limb darkening changes the disc envelope too. Rotating the baseline is necessary to recover an unknown separation direction; a single orientation measures only its projection.