= Rayleigh criterion
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{title2=$\theta_{\rm R}=1.22\lambda/D$}
The <Rayleigh criterion> calls two equal, mutually incoherent <optical point sources> just resolved when one circular-pupil <Airy disk>'s centre coincides with the other's first zero. It is a specified discrimination convention, distinct from <full width at half maximum> and from precision in estimating one source's position.
For this circular <optical pupil>, the <Rayleigh criterion> uses angular separation $s_{\rm R}=\theta_1\simeq1.22\lambda/D$, or $s_{\rm R}=r_1\simeq1.22\lambda f/D$ in the <paraxial approximation> to the <focal plane>. The <point spread functions> still overlap.<Oscillations and Waves by Richard Fitzpatrick/Two-Dimensional Fourier Optics>[equations 10.85–10.87: circular-aperture first-zero scale and Rayleigh convention]{ref}
For the <Airy disk> design example with $D=5.00\,\mathrm{mm}$, $f=100\,\mathrm{mm}$ and $\lambda=550\,\mathrm{nm}$, the angular convention gives $s_{\rm R}=1.34164\times10^{-4}\,\mathrm{rad}$, or about $27.67$ <arcseconds>, and the image-plane separation is $13.4164\,\mathrm{\mu m}$. Two equally bright, mutually incoherent <optical point sources> at separations $0.7$, $1$ and $1.5$ times this scale would have image separations $9.391$, $13.416$ and $20.125\,\mathrm{\mu m}$ respectively. These calculated cases are compared in <incoherent two-source Airy profile>.
The <Rayleigh criterion> is useful because it ties an easily recognized two-peak convention to a predictable optical scale. It does not assert that all information vanishes below that separation. Inferring two unknown positions also depends on <photon shot noise>, background, unequal brightness and uncertainty in the <point spread function>; locating a single isolated source is a different task. A theoretical analysis shows that alternative optical measurements can estimate the separation of two incoherent <optical point sources> below this scale.<Quantum Theory of Superresolution for Two Incoherent Optical Point Sources by Mankei Tsang and others>[abstract: linear optics and photon counting can estimate sub-Rayleigh separation]{ref}
The mutual <optical coherence> condition is essential. In <incoherent imaging>, time-averaged <light intensities> add. In <coherent imaging>, <wave amplitudes> add before squaring, leaving a <wave interference> term controlled by relative <wave phase>. Two holes illuminated by the same coherent <laser> therefore do not implement the equal-incoherent-source convention merely because their individual patterns resemble <Airy disks>. The coherent single-aperture laboratory test establishes the response of each aperture; it does not establish the pair's mutual incoherence.
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