Van Cittert–Zernike theorem

Remarks

Proof idea: the field at each observation point is a superposition of the source field, propagated by the Huygens–Fresnel principle. The mutual intensity at $P_1, P_2$ is then a double integral over pairs of source points. Spatial incoherence makes the mutual intensity of the source proportional to $I(\xi, \eta)$ times a delta function on the diagonal, so the double integral collapses to a single one. In the paraxial approximation the remaining phase is linear in $(\xi, \eta)$ up to terms that depend only on $P_1$ and $P_2$, which gives a Fourier transform.

Example: a uniform circular source of radius $a$ gives $\abs{\mu_{12}} = \abs{2 J_1(v) / v}$ with $v = 2\pi a d / (\bar\lambda z)$ and $d = \abs{P_1 P_2}$, where $J_1$ is the Bessel function of order $1$. The first zero of $J_1$ at $v \approx 3.832$ gives the distance $d \approx 0.61 \, \bar\lambda z / a$ at which the coherence first vanishes. This is how the Michelson stellar interferometer measures the angular diameter $2a/z$ of a star.

Without spatial incoherence the result fails: a laser beam across the source plane is coherent everywhere, whatever its intensity profile.

Going deeper

For a student

The theorem has the same mathematical form as Fraunhofer diffraction, with the source intensity in the role of the aperture. This is why the coherence pattern of a uniform disc looks like the Airy pattern of a circular hole. In radio astronomy the theorem is the basis of aperture synthesis. Correlating the signals of pairs of antennas samples the Fourier transform of the sky brightness, which is then inverted to form an image. In optics it explains why light from an extended thermal source becomes partially coherent after propagation, and how large a source may be for a given coherence area. Its hypotheses are quasi-monochromatic light and small path differences; for broadband light, one applies it frequency by frequency.