View source for Angular resolution

{{short description|Ability of any image-forming device to distinguish small details of an object}}
{{about|optics and imaging systems|angular resolution in graph drawing|angular resolution (graph drawing)}}
{{More citations needed|date=January 2012}}
[[File:Event Horizon Telescope and Apollo 16.png|thumb|A series of images representing the magnification of [[M87*]] with an [[angular size]] of some [[microarcsecond]]s, comparable to viewing a tennis ball on the Moon (magnification from top left corner counter−clockwise to the top right corner).]]
'''Angular resolution''' describes the ability of any [[image-forming device]] such as an [[Optical telescope|optical]] or [[radio telescope]], a [[microscope]], a [[camera]], or an [[Human eye|eye]], to distinguish small details of an object, thereby making it a major determinant of [[image resolution]].  It is used in [[optics]] applied to light waves, in [[antenna (radio)|antenna theory]] applied to radio waves, and in [[acoustics]] applied to sound waves.  The colloquial use of the term "resolution" sometimes causes confusion; when an optical system is said to have a high resolution or high angular resolution, it means that the perceived distance, or actual angular distance, between resolved neighboring objects is small. The value that quantifies this property, ''θ,'' which is given by the Rayleigh criterion, is low for a system with a high resolution. The closely related term [[spatial resolution]] refers to the precision of a measurement with respect to space, which is directly connected to angular resolution in imaging instruments.  The '''Rayleigh criterion''' shows that the minimum angular spread that can be resolved by an image-forming system is limited by [[diffraction]] to the ratio of the [[wavelength]] of the waves to the [[aperture]] width.  For this reason, high-resolution imaging systems such as astronomical [[telescope]]s, long distance [[telephoto lens|telephoto camera lenses]] and [[radio telescope]]s have large
==Definition of terms==

''Resolving power'' is the ability of an imaging device to separate (i.e., to see as distinct) points of an object that are located at a small [[angular separation|angular distance]] or it is the power of an optical instrument to separate far away objects, that are close together, into individual images. The term ''[[Optical resolution|resolution]]'' or ''minimum resolvable distance'' is the minimum distance between distinguishable objects in an image, although the term is loosely used by many users of microscopes and telescopes to describe resolving power. As explained below, diffraction-limited resolution is defined by the Rayleigh criterion as the angular separation of two point sources when the maximum of each source lies in the first minimum of the diffraction pattern ([[Airy disk]]) of the other.  In scientific analysis, in general, the term "resolution" is used to describe the [[Accuracy and precision|precision]] with which any instrument measures and records (in an image or spectrum) any variable in the specimen or sample under study.

==The Rayleigh criterion==
{{distinguish|Rayleigh roughness 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>
{{cite book
 |last1=Born |first1=M. |author-link=Max Born
 |last2=Wolf |first2=E. |author2-link=Emil Wolf
 |date=1999
 |title=[[Principles of Optics]]|page=[https://archive.org/details/principlesoptics00born/page/n496 461]
 |publisher=[[Cambridge University Press]]
 |isbn=0-521-64222-1
}}</ref><ref name=rayleigy1879>
{{cite journal
 |last=Lord Rayleigh |first=F.R.S.
 |date=1879
 |title=Investigations in optics, with special reference to the spectroscope
 |journal=[[Philosophical Magazine]]
 |series=5 |volume=8 |issue=49 |pages=261–274
 |doi=10.1080/14786447908639684
|url=https://zenodo.org/record/1431143
 }}</ref> as shown in the accompanying photos. (In the bottom photo on the right that shows the Rayleigh criterion limit, the central maximum of one point source might look as though it lies outside the first minimum of the other, but examination with a ruler verifies that the two do intersect.) If the distance is greater, the two points are well resolved and if it is smaller, they are regarded as not resolved. Rayleigh defended this criterion on sources of equal strength.<ref name=rayleigy1879 />

Considering diffraction through a circular aperture, this translates into:
:<math> \theta\approx 1.22 \frac{\lambda}{D}\quad(\text{considering that}\,\sin\theta\approx\theta)</math>
where ''θ'' is the ''angular resolution'' ([[radians]]), ''λ'' is the [[wavelength]] of light, and ''D'' is the [[diameter]] of the lens' aperture. The factor 1.22 is derived from a calculation of the position of the first dark circular ring surrounding the central [[Airy disc]] of the [[diffraction]] pattern. This number is more precisely 1.21966989... ({{OEIS2C|A245461}}), the first zero of the order-one [[Bessel function of the first kind]] <math>J_{1}(x)</math> divided by [[pi|π]].

The formal Rayleigh criterion is close to the [[empirical]] resolution limit found earlier by the English astronomer [[W. R. Dawes]], who tested human observers on close binary stars of equal brightness. The result, ''θ'' = 4.56/''D'', with ''D'' in inches and ''θ'' in [[arcsecond]]s, is slightly narrower than calculated with the Rayleigh criterion. A calculation using Airy discs as point spread function shows that at [[Dawes' limit]]<!-- that stub may redirect here --> there is a 5% dip between the two maxima, whereas at Rayleigh's criterion there is a 26.3% dip.<ref name="Michalet2006">
{{cite journal
 |last1=Michalet |first1=X.
 |year=2006
 |title=Using photon statistics to boost microscopy resolution
 |journal=[[Proceedings of the National Academy of Sciences]]
 |volume=103 |issue=13 |pages=4797–4798
 |bibcode=2006PNAS..103.4797M
 |doi=10.1073/pnas.0600808103
 |pmid=16549771
 |pmc=1458746
|doi-access=free
 }}</ref> Modern [[image processing]] techniques including [[deconvolution]] of the point spread function allow resolution of binaries with even less angular separation.

Using a  [[small-angle approximation]], the angular resolution may be converted into a ''[[spatial resolution]]'', Δ''ℓ'', by multiplication of the angle (in radians) with the distance to the object. For a microscope, that distance is close to the [[focal length]] ''f'' of the [[Objective (optics)|objective]]. For this case, the Rayleigh criterion reads:
:<math> \Delta \ell \approx 1.22 \frac{ f \lambda}{D}</math>.

This is the [[radius]], in the imaging plane, of the smallest spot to which a [[collimated]] beam of [[light]] can be focused, which also corresponds to the size of the smallest object that the lens can resolve.<ref>
{{cite web
 |date=2002 
 |title=Diffraction: Fraunhofer Diffraction at a Circular Aperture 
 |url=https://www.cvimellesgriot.com/products/Documents/TechnicalGuide/fundamental-Optics.pdf 
 |work=Melles Griot Optics Guide 
 |publisher=[[Melles Griot]] 
 |access-date=2011-07-04 
 |url-status=dead 
 |archive-url=https://web.archive.org/web/20110708214325/http://www.cvimellesgriot.com/products/Documents/TechnicalGuide/fundamental-Optics.pdf 
 |archive-date=2011-07-08 
}}</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>

==Specific cases==
[[Image:Diffraction limit diameter vs angular resolution.svg|thumb|Log–log plot of aperture diameter vs angular resolution at the diffraction limit for various light wavelengths compared with various astronomical instruments. For example, the blue star shows that the [[Hubble Space Telescope]] is almost diffraction-limited in the visible spectrum at 0.1 arcsecs, whereas the red circle shows that the human eye should have a resolving power of 20 arcsecs in theory, though normally only 60 arcsecs.]]

===Single telescope===

Point-like sources separated by an [[angle]] smaller than the angular resolution cannot be resolved. A single optical telescope may have an angular resolution less than one [[arcsecond]], but [[astronomical seeing]] and other atmospheric effects make attaining this very hard.

The angular resolution ''R'' of a telescope can usually be approximated by
:<math>R = \frac {\lambda}{D} </math>
where ''λ'' is the [[wavelength]] of the observed radiation, and ''D'' is the diameter of the telescope's [[Objective (optics)|objective]]. The resulting ''R'' is in [[radian]]s. For example, in the case of yellow light with a wavelength of 580&nbsp;[[nanometer|nm]], for a resolution of 0.1 arc second, we need D=1.2&nbsp;m. Sources larger than the angular resolution are called extended sources or diffuse sources, and smaller sources are called point sources.

This formula, for light with a wavelength of about 562&nbsp;nm, is also called the [[Dawes' limit]].

===Telescope array===
The highest angular resolutions for telescopes can be achieved by arrays of telescopes called [[astronomical interferometer]]s: These instruments can achieve angular resolutions of 0.001&nbsp;arcsecond at optical wavelengths, and much higher resolutions at x-ray wavelengths. In order to perform [[aperture synthesis|aperture synthesis imaging]], a large number of telescopes are required laid out in a 2-dimensional arrangement with a dimensional precision better than a fraction (0.25x) of the required image resolution.

The angular resolution ''R'' of an interferometer array can usually be approximated by
:<math>R=\frac {\lambda}{B} </math>
where ''λ'' is the [[wavelength]] of the observed radiation, and ''B'' is the length of the maximum physical separation of the telescopes in the array, called the [[baseline (interferometry)|baseline]]. The resulting ''R'' is in [[radian]]s. Sources larger than the angular resolution are called extended sources or diffuse sources, and smaller sources are called point sources.

For example, in order to form an image in yellow light with a wavelength of 580&nbsp;nm, for a resolution of 1&nbsp;milli-arcsecond, we need telescopes laid out in an array that is 120&nbsp;m&nbsp;×&nbsp;120&nbsp;m with a dimensional precision better than 145&nbsp;nm.

===Microscope===
The resolution ''R'' (here measured as a distance, not to be confused with the angular resolution of a previous subsection) depends on the [[angular aperture]] <math>\alpha</math>:<ref>
{{cite web
 |last1=Davidson |first1=M. W.
 |title=Resolution
 |url=https://www.microscopyu.com/microscopy-basics/resolution
 |website=Nikon’s MicroscopyU
 |publisher=[[Nikon]]
 |access-date=2017-02-01
}}</ref>

:<math>R=\frac{1.22\lambda}{\mathrm{NA}_\text{condenser} + \mathrm{NA}_\text{objective}}</math> where <math>\mathrm{NA}=n\sin\theta</math>.

Here NA is the [[numerical aperture]], <math>\theta</math> is half the included angle <math>\alpha</math> of the lens, which depends on the diameter of the lens and its focal length, <math>n</math> is the [[refractive index]] of the medium between the lens and the specimen, and <math>\lambda</math> is the wavelength of light illuminating or emanating from (in the case of fluorescence microscopy) the sample.

It follows that the NAs of both the objective and the condenser should be as high as possible for maximum resolution. In the case that both NAs are the same, the equation may be reduced to:
:<math>R=\frac{0.61\lambda}{\mathrm{NA}}\approx\frac{\lambda}{2\mathrm{NA}}</math>

The practical limit for <math>\theta</math> is about 70°. In a dry objective or condenser, this gives a maximum NA of 0.95. In a high-resolution [[oil immersion objective|oil immersion lens]], the maximum NA is typically 1.45, when using immersion oil with a refractive index of 1.52. Due to these limitations, the resolution limit of a light microscope using [[visible light]] is about 200&nbsp;[[nanometer|nm]]. Given that the shortest wavelength of visible light is [[Violet (color)|violet]] (<math>\lambda \approx 400\,\mathrm{nm}</math>),

:<math>R=\frac{1.22 \times 400\,\mbox{nm}}{1.45\ +\ 0.95}=203\,\mbox{nm}</math>

which is near 200&nbsp;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&nbsp;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&nbsp;nm have been resolved with both techniques.<ref name=pohl>
{{cite journal
 |last1=Pohl |first1=D. W.
 |last2=Denk |first2=W.
 |last3=Lanz |first3=M.
 |year=1984
 |title=Optical stethoscopy: Image recording with resolution λ/20
 |journal=[[Applied Physics Letters]]
 |volume=44 |issue=7 |page=651
 |bibcode=1984ApPhL..44..651P
 |doi=10.1063/1.94865
|doi-access=free
 }}</ref><ref>
{{cite web
 |last=Dyba |first=M.
 |title=4Pi-STED-Microscopy...
 |url=http://www.mpibpc.mpg.de/groups/hell/4Pi-STED.htm
 |publisher=[[Max Planck Society]], Department of NanoBiophotonics
 |access-date=2017-02-01
}}</ref> In addition to this [[Photoactivated localization microscopy]] can resolve structures of that size, but is also able to give information in z-direction (3D).

==List of telescopes and arrays by angular resolution==
<!---This is a basic list for expansion, before being moved to the dedicated article, which allready exists as a redirect here--->

{| class="wikitable sortable" style="margin: 1em auto 1em auto; width:90%; font-size:95%;"
|-
! Name || class="unsortable"| Image ||data-sort-type="number"| Angular resolution ([[arc seconds]])|| [[Wavelength]] || Type || Site || Year
|-
| [[Global mm-VLBI Array]] (successor to the ''Coordinated Millimeter VLBI Array'')|| || 0.000012 (12 μas)|| radio (at 1.3&nbsp;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 - 
|-
| [[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 -
|-
| [[Hubble Space Telescope]]|| [[File:HST.jpg|50px]] || 0.04 || light (near 500&nbsp;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 -
|-
| [[James Webb Space Telescope]]|| [[File:JWST.jpg|50px]] || 0.1<ref name="Dalcanton Seager Aigrain Battel p. ">{{cite arXiv | last1=Dalcanton | first1=Julianne | last2=Seager | first2=Sara | last3=Aigrain | first3=Suzanne | last4=Battel | first4=Steve | last5=Brandt | first5=Niel | last6=Conroy | first6=Charlie | last7=Feinberg | first7=Lee | last8=Gezari | first8=Suvi | last9=Guyon | first9=Olivier | last10=Harris | first10=Walt | last11=Hirata | first11=Chris | last12=Mather | first12=John | last13=Postman | first13=Marc | last14=Redding | first14=Dave | last15=Schiminovich | first15=David | last16=Stahl | first16=H. Philip | last17=Tumlinson | first17=Jason | title=From Cosmic Birth to Living Earths: The Future of UVOIR Space Astronomy | year=2015 | eprint=1507.04779 | page=| class=astro-ph.IM }}</ref> || infrared (at 2000&nbsp;nm)<ref name="jwst.nasa.gov 2002">{{cite web | title=FAQ Full General Public Webb Telescope/NASA | website=jwst.nasa.gov | date=2002-09-10 | url=https://www.jwst.nasa.gov/content/about/faqs/faq.html | access-date=2022-09-27}}</ref> || [[space telescope]] || [[List of objects at Lagrange points|Sun–Earth L2]] || 2022 -
|}

==See also==
* [[Angular diameter]]
* [[Beam diameter]]
* [[Dawes' limit]]
* [[Diffraction-limited system]]
* [[Ground sample distance]]
* [[Image resolution]]
* [[Optical resolution]]
* [[Sparrow's resolution limit]]
* [[Visual acuity]]

==Notes==
{{reflist|group=Note}}

== References ==
{{reflist}}

==External links==
* [http://www.microscopyu.com/articles/formulas/formulasresolution.html "Concepts and Formulas in Microscopy: Resolution"] by Michael W. Davidson, ''Nikon MicroscopyU'' (website).

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