Spatial coherence encodes angular source size

Place two receiving apertures a transverse distance b apart. Radiation from a source direction θ reaches them with an approximate path difference bθ for small angles. Each source element therefore contributes a definite relative phase, even though its absolute phase fluctuates. Distinct source elements have vanishing mutual correlation in the spatially incoherent source model, so their cross terms disappear after averaging. Their individual contributions to the receiver correlation remain and add with direction-dependent phase.

Assume quasi-monochromatic stationary scalar radiation of wavelength λ, a distant source, small angular extent, and receiver geometry in the far-field or equivalent Fraunhofer regime. Match temporal paths sufficiently well that bandwidth averaging is negligible. These conditions allow brightness to depend only on direction and the phase to be linear in baseline times angle. Nearby sources with important wavefront curvature need a more general propagation calculation; the simple angular transform cannot silently be reused there.

Let the brightness be constant across a full angular width α centered on the axis and zero outside it. Integrating the complex exponential from minus α over two to plus α over two gives a sine divided by its argument. The normalization makes the coherence equal to one at zero baseline. The first zero occurs when the two edge directions differ in phase by a complete cycle, so contributions across the strip cancel in the average.

The result explains the inverse size relation: a smaller angular source stays coherent across a larger receiver separation. It also shows why coherence need not decrease monotonically. A sharp-edged strip has sidelobes beyond its first null, with alternating signs of the real coherence. Visibility measures the magnitude and loses that sign unless fringe phase is also measured. A smooth Gaussian angular profile instead produces a smooth Gaussian baseline dependence, without these sharp-boundary zeros.

A single visibility value generally does not determine an arbitrary source shape. Many brightness distributions can share one Fourier sample, and visibility magnitude alone leaves further phase ambiguities. Measuring several baselines, orientations, and where possible complex correlations supplies more information. Atmospheric phase errors, finite aperture averaging, and spectral bandwidth can alter the observations. The theorem provides the ideal forward relation; calibrating an instrument and reconstructing a source are additional inference problems.

Use the two-wave lab to examine the cancellation underlying the integral. Reset and set both controls to phase zero degrees and ratio one; predict the largest sum. For another fresh comparison Reset and set phase 180 degrees and ratio one; predict zero. These are two coherent contributions at one fixed frequency, not an extended incoherent source. The lab can test phasor cancellation between two terms but cannot vary a baseline, integrate angular brightness, or demonstrate the theorem by itself.