Editing Angular resolution (section)

==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).