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==The Rayleigh criterion==
{{
[[File:Airy disk spacing near Rayleigh criterion.png|thumb|right|[[Airy disk|Airy diffraction patterns]] generated by light from two [[point source]]s passing through a circular [[aperture]], such as the [[pupil]] of the eye. Points far apart (top) or meeting the Rayleigh criterion (middle) can be distinguished. Points closer than the Rayleigh criterion (bottom) are difficult to distinguish.]]
The imaging system's resolution can be limited either by [[optical aberration|aberration]] or by [[diffraction]] causing [[Focus (optics)|blurring]] of the image. These two phenomena have different origins and are unrelated. Aberrations can be explained by [[geometrical optics]] and can in principle be solved by increasing the optical quality of the system. On the other hand, diffraction comes from the wave nature of light and is determined by the finite aperture of the optical elements. The [[lens (optics)|lens]]' circular [[aperture]] is analogous to a two-dimensional version of the [[Slit experiment|single-slit experiment]]. [[Light]] passing through the lens [[Interference (wave propagation)|interferes]] with itself creating a ring-shape diffraction pattern, known as the [[Airy pattern]], if the [[wavefront]] of the transmitted light is taken to be spherical or plane over the exit aperture.
The interplay between diffraction and aberration can be characterised by the [[point spread function]]<!--Maybe should go after--> (PSF). The narrower the aperture of a lens the more likely the PSF is dominated by diffraction. In that case, the angular resolution of an optical system can be estimated (from the [[diameter]] of the aperture and the [[wavelength]] of the light) by the Rayleigh criterion defined by [[Lord Rayleigh]]: two point sources are regarded as just resolved when the principal diffraction maximum (center) of the [[Airy disk]] of one image coincides with the first minimum of the [[Airy disk]] of the other,<ref>
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}}</ref> The size is proportional to wavelength, ''λ'', and thus, for example, [[blue]] light can be focused to a smaller spot than [[red]] light. If the lens is focusing a beam of [[light]] with a finite extent (e.g., a [[laser]] beam), the value of ''D'' corresponds to the [[diameter]] of the light beam, not the lens.{{refn|group=Note|name=GaussianNote|In the case of laser beams, a [[Gaussian beam|Gaussian Optics]] analysis is more appropriate than the Rayleigh criterion, and may reveal a smaller diffraction-limited spot size than that indicated by the formula above.}} Since the spatial resolution is inversely proportional to ''D'', this leads to the slightly surprising result that a wide beam of light may be focused on a smaller spot than a narrow one. This result is related to the [[Fourier uncertainty principle|Fourier properties]] of a lens.
A similar result holds for a small sensor imaging a subject at infinity: The angular resolution can be converted to a spatial resolution on the sensor by using ''f'' as the distance to the [[image sensor]]; this relates the spatial resolution of the image to the [[f-number]], {{f/}}#:
:<math> \Delta \ell \approx 1.22 \frac{f \lambda}{D}=1.22 \lambda \cdot (f/\#)</math>.
Since this is the radius of the Airy disk, the resolution is better estimated by the diameter, <math> 2.44 \lambda \cdot (f/\#)</math>
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This formula, for light with a wavelength of about 562 nm, is also called the [[Dawes' limit]].
One unit for angular resolution in this case of Gaussian dispersion from astronomical seeing is the ''[[half-power diameter]]'', which is the angular diameter in which half of the power from a telescope is centered.<ref>{{Cite web |title=5. X-Ray Telescopes (XRTs) |url=https://www.astro.isas.jaxa.jp/suzaku/research/proposal/ao1_obsolete/astroe2_td/node8.html#SECTION00822000000000000000 |access-date=2025-07-09 |website=www.astro.isas.jaxa.jp}}</ref>
===Telescope array===
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which is near 200 nm.
Oil immersion objectives can have practical difficulties due to their shallow [[depth of field]] and extremely short working distance, which calls for the use of very thin (0.17 mm) cover slips, or, in an inverted microscope, thin glass-bottomed [[Petri dish]]es.
However, resolution below this theoretical limit can be achieved using [[super-resolution microscopy]]. These include optical near-fields ([[Near-field scanning optical microscope]]) or a diffraction technique called [[4Pi STED microscopy]]. Objects as small as 30 nm have been resolved with both techniques.<ref name=pohl>
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| [[Global mm-VLBI Array]] (successor to the ''Coordinated Millimeter VLBI Array'')|| || 0.000012 (12 μas)|| radio (at 1.3 cm) || [[very long baseline interferometry]] array of different [[radio telescope]]s || a range of locations on Earth and in space<ref name="Max Planck Institute for Radio Astronomy 2022">{{cite web | title=Images at the Highest Angular Resolution in Astronomy | website=Max Planck Institute for Radio Astronomy | date=2022-05-13 | url=https://www.mpifr-bonn.mpg.de/pressreleases/2022/2 | access-date=2022-09-26}}</ref> || 2002 -
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| [[Very Large Telescope]]/[[PIONIER (VLTI)|PIONIER]]|| [[File:Paranal and the Pacific at sunset (dsc4088, retouched, cropped).jpg|50px]] || 0.001 (1 mas)|| light (1-2 [[micrometre]])<ref name="de Zeeuw p. ">{{cite journal | last=de Zeeuw | first=Tim | title=Reaching New Heights in Astronomy - ESO Long Term Perspectives | journal=The Messenger | year=2017 | volume=166 | arxiv=1701.01249 | page=2| bibcode=2016Msngr.166....2D }}</ref> || largest [[optical astronomy|optical]] array of 4 [[reflecting telescope]]s || [[Paranal Observatory]], [[Antofagasta Region]], Chile || 2002/2010 -
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| [[Hubble Space Telescope]]|| [[File:HST.jpg|50px]] || 0.04 || light (near 500 nm)<ref name="NASA 2007">{{cite web | title=Hubble Space Telescope | website=NASA | date=2007-04-09 | url=https://www.nasa.gov/missions/highlights/webcasts/shuttle/sts109/hubble-qa.html#:~:text=In%20visible%20light%20(at%20wavelengths,by%20about%2040%20arc%20seconds. | access-date=2022-09-27}}</ref> || [[space telescope]] || [[Geocentric orbit|Earth orbit]] || 1990 -
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