Astronomical interferometer

An astronomical interferometer combines observations from separated collecting locations to infer angular source structure. Its projected interferometric baseline determines the angular spatial frequency sampled. Amplitude and intensity interferometers measure different coherence observables; a Michelson stellar interferometer combines optical field amplitudes.

A two-element radio interferometer coherently combines signals from two receiving antennas. With equal single-arm wave intensity $I_0$, projected baseline $b$, free-space wavelength $\lambda$ and imposed phase difference $\delta$, its directional response is the displayed formula. It follows by adding the two complex amplitudes and taking the squared modulus. Near the normal to the baseline, the first null is $\lambda /(2b)$ from a maximum. This narrow response repeats in interference fringes, so a long baseline alone does not give an unambiguous image.

For three collectors whose measured pair phases are ${\varphi }_{ij}={\varphi }_{ij}^{\mathrm{source}}+{ϵ}_i-{ϵ}_j$, the oriented sum around the triangle cancels the collector-based errors. closure phase retains source phase information, including asymmetry, but does not cancel arbitrary baseline-dependent errors or fill missing spatial-frequency coverage.

The Michelson stellar interferometer, associated with Albert Abraham Michelson, uses separated collectors and beam-redirection optics to produce interference fringes. The collector spacing determines sensitivity to a stellar angular diameter, independently of the final beam spacing at the combiner. The first null for a uniform disk is $D=1.21967\lambda /\alpha$, where $\alpha$ is the full angular diameter. Different surface-brightness laws, including limb darkening, change the null.

Let identical spatially incoherent stellar brightness profiles of equal integrated flux be displaced by projected angles $\pm \beta /2$ along an interferometer baseline. The Fourier shift property makes their normalized signed coherence $g(Y)=g_{\star }(Y)\cos (\pi \beta Y/\lambda )$. The observed visibility is its absolute value. The slow single-star envelope depends on each star's brightness profile, while the oscillatory factor records projected separation. Unequal flux prevents complete cancellation at the binary minima.

For two identical equally bright stars, zeros of the binary factor occur at $Y_j=(j+1/2)\lambda /\beta$, where $\beta$ is separation projected along the baseline. Their repeated spacing is $\Delta Y=\lambda /\beta$. Multiple samples separate these rapid zeros from zeros of a resolved stellar envelope. Baseline rotation is needed to recover an unknown two-dimensional separation direction; the first zero alone does not determine both diameter and separation.

Two identical uniform circular discs of angular diameter $\alpha$ and equal flux have $g_{\star }(Y)=2J_1(x)/x$, $x=\pi \alpha Y/\lambda$, with limiting value one at $Y=0$. Multiplication by $\cos (\pi \beta Y/\lambda )$ gives their signed binary coherence. Its first single-disc envelope zero is $Y=1.21967\lambda /\alpha$, while the binary modulation has a separate scale $\lambda /\beta$. Limb darkening and other nonuniform profiles change the envelope, not the elementary Fourier shift factor.