The van Cittert-Zernike Coherence Theorem

Derivation

The van Cittert-Zernike Coherence Theorem

D-211 Home PU-206 Threads light · waves · chance Depends on Fraunhofer Diffraction as a Fourier Transform, fourier-transform-pairs

Statement

For a spatially incoherent, quasi-monochromatic extended source of angular intensity distribution \(I(\hat{\mathbf{s}})\), the complex degree of spatial coherence \(\mu_{12}\) between two observation points \(P_1,P_2\) separated by baseline \(\Delta\vec{r}\) in a distant plane equals the normalized Fourier transform of the source intensity distribution, evaluated at the spatial frequency set by the baseline: \[ \mu_{12}=\frac{\displaystyle\int I(\hat{\mathbf{s}})\,e^{-i\,k\,\hat{\mathbf{s}}\cdot\Delta\vec{r}}\,d\Omega}{\displaystyle\int I(\hat{\mathbf{s}})\,d\Omega}. \]

Why it matters

Light from stars, discharge lamps and thermal sources is generated by an enormous number of independently radiating atoms; each atom's emission bears no fixed phase relation to any other. Yet by the time this light reaches us, spatial correlations have appeared: two nearby points on a wavefront can vibrate in near-lockstep even though the source was chaotic. The van Cittert-Zernike theorem quantifies exactly how much correlation survives, and it turns out to encode a complete Fourier image of the source.

This is the theoretical foundation of stellar interferometry (Michelson, and modern arrays like VLTI and CHARA), of aperture synthesis in radio astronomy, and of coherence-based image reconstruction. It explains why a mercury lamp a few metres away produces beautifully visible interference fringes in a double slit despite being "incoherent" at the source: distance itself manufactures coherence.

Assumptions

Spatial incoherence at the source.Distinct source elements radiate with mutually uncorrelated, statistically independent fields, \(\langle E(\hat{\mathbf{s}}_1)E^{*}(\hat{\mathbf{s}}_2)\rangle \propto I(\hat{\mathbf{s}}_1)\,\delta(\hat{\mathbf{s}}_1-\hat{\mathbf{s}}_2)\). If dropped, cross-terms between source points survive and the coherence is no longer simply the transform of the intensity.

Quasi-monochromatic light.The bandwidth is narrow, \(\Delta\nu \ll \bar{\nu}\), so a single wavenumber \(k=2\pi/\bar{\lambda}\) governs all propagation phases. If dropped, temporal-coherence effects mix in and the mutual coherence function acquires an explicit time-delay dependence.

Far-field (Fraunhofer) geometry with small source.The source-to-aperture distance \(R\) greatly exceeds both source size and baseline, so spherical wavefronts are locally planar and quadratic phase terms cancel between the two points. If dropped, an extra Fresnel phase \(\exp[i k(r_1^2-r_2^2)/2R]\) multiplies \(\mu_{12}\) and the clean transform relation is spoiled.

Scalar, unpolarized, stationary field.Polarization is ignored and the statistics are time-stationary and ergodic, so ensemble averages equal time averages. If dropped, one needs the full coherence tensor and non-stationary corrections.

Derivation

\[ \Gamma_{12}=\langle E(P_1,t)\,E^{*}(P_2,t)\rangle \]

Definition of the mutual (equal-time) coherence function between the two observation points; the correlation of the analytic signals at \(P_1\) and \(P_2\). A

\[ E(P,t)=\int_{\text{source}} K\,\frac{A(\hat{\mathbf{s}})}{r_P(\hat{\mathbf{s}})}\,e^{i\left(k\,r_P(\hat{\mathbf{s}})-\omega t\right)}\,dS \]

Huygens–Fresnel superposition: the field at \(P\) is the coherent sum of spherical wavelets from every source element \(dS\), with amplitude \(A(\hat{\mathbf{s}})\), obliquity constant \(K\), and \(r_P\) the distance from that element to \(P\). A

\[ \Gamma_{12}=\iint \frac{K^2}{r_1 r_2}\,\big\langle A(\hat{\mathbf{s}})A^{*}(\hat{\mathbf{s}}')\big\rangle\,e^{i k\left(r_1(\hat{\mathbf{s}})-r_2(\hat{\mathbf{s}}')\right)}\,dS\,dS' \]

Insert the superposition into \(\Gamma_{12}\); the time factor \(e^{-i\omega t}\) cancels against its conjugate. \(r_1,r_2\) denote distances from element \(\hat{\mathbf{s}}\) to \(P_1\) and \(\hat{\mathbf{s}}'\) to \(P_2\). B

\[ \big\langle A(\hat{\mathbf{s}})A^{*}(\hat{\mathbf{s}}')\big\rangle = I(\hat{\mathbf{s}})\,\delta(\hat{\mathbf{s}}-\hat{\mathbf{s}}') \]

Apply spatial incoherence: independent source elements have delta-correlated amplitudes, with \(I(\hat{\mathbf{s}})=\langle|A|^2\rangle\) the intensity. This collapses the double integral to a single one. C

\[ \Gamma_{12}=\int \frac{K^2}{r_1 r_2}\,I(\hat{\mathbf{s}})\,e^{i k\left(r_1(\hat{\mathbf{s}})-r_2(\hat{\mathbf{s}})\right)}\,dS \]

Perform the \(\hat{\mathbf{s}}'\) integration against the delta function; only the diagonal \(\hat{\mathbf{s}}'=\hat{\mathbf{s}}\) contributes, so a single source integral remains. B

\[ r_1-r_2 \;\approx\; -\,\hat{\mathbf{s}}\cdot(\vec{r}_1-\vec{r}_2)\;=\;-\,\hat{\mathbf{s}}\cdot\Delta\vec{r} \]

Far-field expansion: for a source element in direction \(\hat{\mathbf{s}}\), the path-length difference to the two observation points is the projection of the baseline \(\Delta\vec{r}=\vec{r}_1-\vec{r}_2\) onto \(\hat{\mathbf{s}}\); quadratic (Fresnel) terms cancel between \(r_1\) and \(r_2\) at this order. C

\[ \Gamma_{12}\;\propto\;\int I(\hat{\mathbf{s}})\,e^{-i\,k\,\hat{\mathbf{s}}\cdot\Delta\vec{r}}\,d\Omega \]

Substitute the far-field phase and absorb the slowly varying prefactor \(K^2/r_1 r_2\) (nearly constant across the small source) and the element-of-source-to-solid-angle Jacobian \(dS\to R^2\,d\Omega\) into the proportionality constant. This is a Fourier transform of \(I\) against the baseline. B

\[ \mu_{12}\equiv\frac{\Gamma_{12}}{\sqrt{\Gamma_{11}\,\Gamma_{22}}}=\frac{\Gamma_{12}}{I_0} \]

Normalize by the self-coherences (the intensities \(\Gamma_{11}=\Gamma_{22}=I_0=\int I\,d\Omega\)) to form the complex degree of coherence, bounded by \(|\mu_{12}|\le 1\). For \(\Delta\vec{r}=0\) the exponential is unity and \(\mu=1\), fixing the normalization. B

\[ \boxed{\;\mu_{12}=\dfrac{\displaystyle\int I(\hat{\mathbf{s}})\,e^{-i\,k\,\hat{\mathbf{s}}\cdot\Delta\vec{r}}\,d\Omega}{\displaystyle\int I(\hat{\mathbf{s}})\,d\Omega}\;} \]

Combine steps 7 and 8: the constant prefactor cancels in the ratio, leaving the normalized transform. The structural identity with Fraunhofer diffraction as a Fourier transform is exact — the source intensity plays the role of the aperture transmission. A

Result

\[ \mu_{12}=\frac{\displaystyle\int I(\hat{\mathbf{s}})\,e^{-i\,k\,\hat{\mathbf{s}}\cdot\Delta\vec{r}}\,d\Omega}{\displaystyle\int I(\hat{\mathbf{s}})\,d\Omega} \]

Reading. The correlation between the light at two points is set entirely by the source's shape on the sky, not its brightness. Sweeping the baseline \(\Delta\vec{r}\) samples different spatial frequencies of the source; measuring \(\mu_{12}\) at many baselines and inverting the transform reconstructs \(I(\hat{\mathbf{s}})\). A point source (\(I\propto\delta\)) gives \(\mu_{12}=1\) for all baselines — perfect coherence — while a broad source decorrelates quickly, so the "coherence area" at the observation plane is inversely proportional to the source's angular size.

Units check. The exponent \(k\,\hat{\mathbf{s}}\cdot\Delta\vec{r}\) has units \((\text{m}^{-1})(\text{dimensionless})(\text{m})=\) dimensionless, as a phase must be. Numerator and denominator both carry units of \([I]\cdot\text{sr}\), so their ratio \(\mu_{12}\) is dimensionless, consistent with \(|\mu_{12}|\le 1\).

Limiting cases

Point source: \(I(\hat{\mathbf{s}})=I_0\,\delta(\hat{\mathbf{s}}-\hat{\mathbf{s}}_0)\Rightarrow \mu_{12}=e^{-ik\hat{\mathbf{s}}_0\cdot\Delta\vec{r}}\), so \(|\mu_{12}|=1\): fully coherent at every separation.
Zero baseline: \(\Delta\vec{r}=0\Rightarrow \mu_{12}=1\), the field is perfectly self-correlated at a single point.
Uniform circular disc (angular radius \(\alpha\)): \(|\mu_{12}|=\left|\dfrac{2J_1(x)}{x}\right|\), \(x=k\alpha\,|\Delta\vec{r}|\); first zero at \(|\Delta\vec{r}|=0.61\,\lambda/\alpha\) — the Michelson stellar-diameter formula.
Uniform slit source of angular width \(\theta_s\): \(|\mu_{12}|=|\operatorname{sinc}(\theta_s\,\Delta r/\lambda)|\), first null at \(\Delta r=\lambda/\theta_s\).
Very large source (\(\theta_s\to\infty\)): coherence area shrinks to zero; adjacent points are uncorrelated, recovering the incoherent limit.

Breaks when

Partially coherent or laser-like sources. If source elements are already correlated (e.g. a laser, or a source viewed through a coherent relay), the delta-correlation of step 4 fails; cross-terms survive and \(\mu_{12}\) is no longer the plain transform of \(I\).
Near field / Fresnel regime. When \(R\) is not large compared with source size or baseline, the quadratic path terms of step 6 do not cancel. An extra Fresnel phase factor \(\exp\!\big[i k(|\vec{r}_1|^2-|\vec{r}_2|^2)/2R\big]\) multiplies the transform, and one must use the generalized (Fresnel-form) van Cittert-Zernike relation.
Broadband light. For large \(\Delta\nu/\bar\nu\), the single-\(k\) assumption collapses; temporal coherence enters and the full space-time mutual coherence function \(\Gamma_{12}(\tau)\) is required, not the quasi-monochromatic \(\mu_{12}\).
Optically thick or scattering intervening medium. Turbulence or scattering between source and aperture imprints additional random phases, degrading \(|\mu_{12}|\) below the theorem's prediction (the basis of seeing-limited resolution).

Failure modes

Confusing spatial and temporal coherence. The theorem addresses correlation between two points (spatial), governed by source angular size; it says nothing about correlation between two times, which is governed by bandwidth. Students routinely conflate the coherence area with the coherence length.
Forgetting the normalization. Writing \(\mu_{12}\) as the raw transform \(\int I e^{-ik\hat{\mathbf{s}}\cdot\Delta\vec{r}}d\Omega\) without dividing by \(\int I\,d\Omega\) gives a quantity with the wrong units and \(|\mu|\) exceeding 1 at zero baseline.
Sign / direction of the transform. Dropping the minus sign in the exponent, or using \(\Delta\vec{r}\) instead of the baseline projected on \(\hat{\mathbf{s}}\), flips the conjugate and can invert reconstructed images.
Treating brightness as relevant. Assuming a brighter source is "more coherent." Multiplying \(I\) by a constant leaves \(\mu_{12}\) unchanged — only the shape matters.
Using \(2\pi\) inconsistently. Mixing \(k=2\pi/\lambda\) conventions with a \(1/\lambda\) spatial-frequency convention, producing factors of \(2\pi\) errors in the fringe-visibility null spacing.

Discussion

The deep content of the theorem is that propagation manufactures coherence. At the source, the field is delta-correlated: neighbouring atoms are statistically independent. But each observation point receives contributions from the whole source, and after propagating a distance \(R\) the phase relationships between source elements become smooth functions of position in the observation plane. Two nearby observation points see almost the same superposition, hence they are correlated. The larger the distance and the smaller the source, the wider this "coherence area." This is why sunlight, from a source subtending half a degree, has a coherence area of only \(\sim\!50\,\mu\text{m}\) at the ground, whereas a distant star produces coherence areas metres across — enough to run a stellar interferometer.

The structural parallel with Fraunhofer diffraction is not a coincidence but a duality. In diffraction, a coherent aperture with transmission \(t(\vec{x})\) produces a far-field amplitude that is the Fourier transform of \(t\). Here an incoherent source with intensity \(I(\hat{\mathbf{s}})\) produces a coherence function that is the Fourier transform of \(I\). The source intensity plays the mathematical role that the aperture field played, with the baseline \(\Delta\vec{r}\) as the conjugate (spatial-frequency) variable. This is why the same Bessel and sinc functions reappear in both problems.

The measurable consequence is fringe visibility. In a Young's double-slit or Michelson stellar interferometer, the fringe contrast \(V\) equals \(|\mu_{12}|\) (for equal intensities at the two apertures). Michelson exploited exactly this in 1920 to measure the angular diameter of Betelgeuse: he widened his interferometer baseline until the fringes vanished (\(|\mu_{12}|\to 0\)), then read off the stellar diameter from the first-null condition \(0.61\,\lambda/\alpha\). Modern aperture synthesis generalizes this: an array measures \(\mu_{12}\) at hundreds of baselines (the "\(uv\)-plane") and Fourier-inverts to make an image — the operating principle of radio interferometers and the Event Horizon Telescope.

At the deepest level the theorem is a statement about the propagation of the two-point correlation function of a random field. The mutual coherence \(\Gamma_{12}\) itself obeys a pair of wave equations (the Wolf equations), one in each spatial argument, and the van Cittert-Zernike result is the far-field asymptotic solution for an incoherent source boundary condition. It is the classical, second-order-correlation ancestor of the quantum Hanbury Brown-Twiss effect, where intensity correlations \(\langle I_1 I_2\rangle\) rather than field correlations are measured; for thermal light the Siegert relation ties the two together as \(\langle I_1 I_2\rangle = \langle I_1\rangle\langle I_2\rangle(1+|\mu_{12}|^2)\), letting intensity interferometry recover \(|\mu_{12}|\) without phase stability.

Common misconceptions. "Incoherent source" does not mean "no coherence anywhere" — it describes the source plane only; coherence emerges downstream. And the theorem gives the complex \(\mu_{12}\): its modulus is fringe visibility, but its phase carries the position/asymmetry information essential for true imaging, which is why phase retrieval (or phase-closure techniques) is central to interferometric astronomy.

Worked examples

\[ \text{Coherence radius of sunlight: } |\Delta\vec r|_{\text{coh}}=0.61\,\frac{\lambda}{\alpha} \]

Model the Sun as a uniform disc of angular radius \(\alpha\). The van Cittert-Zernike coherence follows the \(2J_1(x)/x\) pattern; the first zero (edge of the coherence area) sits at \(x=k\alpha|\Delta\vec r|=3.83\), i.e. \(|\Delta\vec r|=0.61\,\lambda/\alpha\). B

\[ \alpha=\tfrac12(0.53^\circ)=4.6\times10^{-3}\ \text{rad},\qquad \lambda=550\ \text{nm} \]

Insert numbers: solar angular diameter \(0.53^\circ\), so angular radius \(\alpha\approx 4.6\times10^{-3}\) rad; take mid-visible \(\lambda=550\) nm. A

\[ |\Delta\vec r|_{\text{coh}}=0.61\times\frac{5.5\times10^{-7}\ \text{m}}{4.6\times10^{-3}}\approx 7.3\times10^{-5}\ \text{m} \]

Evaluate. The coherence radius is about \(73\ \mu\text{m}\); the coherence diameter is roughly \(0.15\) mm. A

\[ |\Delta\vec r|_{\text{coh}}\approx 73\ \mu\text{m} \]

Reading. Two points on the ground more than about \(0.15\) mm apart see essentially uncorrelated sunlight — which is why you cannot get high-contrast double-slit fringes from the Sun with widely spaced slits, and why direct sunlight makes a poor coherent source.

Units check. \(\lambda/\alpha=\text{m}/\text{rad}=\text{m}\); dimensionless \(0.61\) preserves metres. Correct.

\[ |\Delta\vec r|_{\text{null}}=1.22\,\frac{\lambda}{2\alpha}=0.61\,\frac{\lambda}{\alpha} \]

Michelson stellar interferometer on Betelgeuse: the fringe visibility \(V=|\mu_{12}|=|2J_1(x)/x|\) first vanishes at baseline \(D=|\Delta\vec r|_{\text{null}}=1.22\,\lambda/(2\alpha)\), with \(2\alpha\) the stellar angular diameter. B

\[ 2\alpha=0.047\ \text{arcsec}=2.3\times10^{-7}\ \text{rad},\qquad \lambda=570\ \text{nm} \]

Betelgeuse's measured angular diameter is about \(0.047''\); convert \(0.047\times(4.85\times10^{-6}\ \text{rad/arcsec})=2.3\times10^{-7}\) rad. Michelson worked near \(\lambda=570\) nm. A

\[ D=1.22\,\frac{5.7\times10^{-7}\ \text{m}}{2.3\times10^{-7}}\approx 3.0\ \text{m} \]

Solve for the baseline at which fringes disappear: about \(3\) m. Michelson's beam-combining mirrors on the 100-inch telescope were indeed spread to roughly this separation. A

\[ D_{\text{null}}\approx 3.0\ \text{m} \]

Reading. Widening the interferometer baseline to \(\sim\!3\) m collapses the fringes; reading this null off the \(2J_1/x\) curve yields Betelgeuse's angular diameter without ever resolving the disc in a conventional telescope — a direct application of van Cittert-Zernike.

Units check. \(\lambda/(2\alpha)=\text{m}/\text{rad}=\text{m}\). The result is metres. Correct.

Problems

Coherence from a lamp. A quasi-monochromatic mercury lamp (\(\lambda=546\) nm) is masked to a circular aperture of diameter \(1.0\) mm and viewed at distance \(2.0\) m. Estimate the coherence radius in the viewing plane.
Solution
Angular radius \(\alpha=(0.5\ \text{mm})/(2.0\ \text{m})=2.5\times10^{-4}\) rad. Coherence radius \(|\Delta\vec r|_{\text{coh}}=0.61\,\lambda/\alpha=0.61\times(5.46\times10^{-7})/(2.5\times10^{-4})\approx 1.3\times10^{-3}\) m \(=1.3\) mm. So slits up to \(\sim\!1.3\) mm apart will still show good fringes.
Slit source visibility. A uniform incoherent slit source of angular width \(\theta_s=1.0\times10^{-3}\) rad illuminates two pinholes separated by \(\Delta r\). At \(\lambda=500\) nm, find the pinhole separation giving the first zero of fringe visibility.
Solution
For a uniform strip, \(|\mu_{12}|=|\operatorname{sinc}(\theta_s\Delta r/\lambda)|\), first null when \(\theta_s\Delta r/\lambda=1\), i.e. \(\Delta r=\lambda/\theta_s=(5.0\times10^{-7})/(1.0\times10^{-3})=5.0\times10^{-4}\) m \(=0.50\) mm.
Radio interferometry. A radio array observes at \(\lambda=21\) cm. A source has angular size \(2\times10^{-6}\) rad (treated as a uniform disc, so use \(0.61\,\lambda/\alpha\) with \(\alpha\) the radius \(1\times10^{-6}\) rad). What baseline is needed to reach the first coherence null?
Solution
\(D=0.61\,\lambda/\alpha=0.61\times(0.21)/(1\times10^{-6})=1.28\times10^{5}\) m \(\approx 130\) km. This is why very-long-baseline interferometry (VLBI) is required to resolve compact radio sources.
Complex phase of \(\mu_{12}\). A point source lies at angle \(\hat{\mathbf{s}}_0\) with \(\hat{\mathbf{s}}_0\cdot\Delta\vec{r}=0.30\ \mu\text{m}\); \(\lambda=600\) nm. Compute \(\mu_{12}\) (magnitude and phase).
Solution
For a point source \(\mu_{12}=e^{-ik\hat{\mathbf{s}}_0\cdot\Delta\vec r}\), so \(|\mu_{12}|=1\). Phase \(=-k(0.30\ \mu\text{m})=-(2\pi/0.600\ \mu\text{m})(0.30\ \mu\text{m})=-\pi\) rad. Thus \(\mu_{12}=e^{-i\pi}=-1\): fully coherent but the fringe pattern is shifted by half a period.
Hanbury Brown-Twiss link. For thermal light with \(|\mu_{12}|=0.6\), use the Siegert relation to find the normalized intensity correlation \(g^{(2)}(\Delta\vec r)=\langle I_1 I_2\rangle/(\langle I_1\rangle\langle I_2\rangle)\).
Solution
Siegert: \(g^{(2)}=1+|\mu_{12}|^2=1+(0.6)^2=1+0.36=1.36\). The excess of \(0.36\) above unity is the photon-bunching signal an intensity interferometer measures; taking its square root recovers \(|\mu_{12}|=0.6\), giving the source size without phase-stable optics.