Editing Angular resolution (section)

===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).