Editing Diffraction-limited system (section)

===The Abbe diffraction limit for a microscope===
The observation of sub-wavelength structures with microscopes is difficult because of the '''Abbe diffraction limit'''. [[Ernst Abbe]] first mention the diffraction limit in his 1873 paper, page 466: „[…] die physikalische Unterscheidungsgrenze […] hängt allein vom Oeffnungswinkel ab und ist dem Sinus seines halben Betrages proportional“, or "[…] the physical limit of resolution […] depends solely on the aperture angle and is proportional to the sine of half its magnitude".<ref>{{cite journal|last1=Abbe|first1=Ernst|title=Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung|journal=Archiv für mikroskopische Anatomie|date=1873|volume=9|pages=413–468|doi=10.1007/BF02956173 |url=https://doi.org/10.1007/BF02956173}}</ref> Abbe wrote it in form of a formula in his 1882 paper, page 461: "The smallest dimensions which are within the reach of a given aperture are indicated with sufficient accuracy by taking the limit of the resolving or separating power of that aperture for periodic or regular structures, i.e. the minimum distance apart at which given elements can be delineated separately with the aperture in question. The numerical expression of that minimum distance is" <ref>{{cite journal|last1=Abbe|first1=Ernst|title=The Relation of Aperture and Power in the Microscope (continued)|journal=Journal of the Royal Microscopical Society|date=1882|volume=2|issue=4 |pages=460–473|doi=10.1111/j.1365-2818.1882.tb04805.x |url=https://doi.org/10.1111/j.1365-2818.1882.tb04805.x|url-access=subscription}}</ref> 
:<math>d=\frac{ \lambda}{2 n \sin \theta} = \frac{\lambda}{2\mathrm{NA}}</math>,
where <math>\lambda</math> is the wavelength, <math>n</math> is the refractive index of the medium, and <math>\theta</math> is the semi-angle of the light focused by the optical system. The same formula had been proven by Hermann von Helmholtz in 1874.<ref>{{cite journal|last1=von Helmholtz|first1=Hermann|title=Die theoretische Grenze für die Leistungsfähigkeit der Mikroskope|trans-title=The Theoretical Limit of the Efficiency of Microscopes)|journal=Annalen der Physik und Chemie: Jubelband dem Herausgeber Johann Christian Poggendorff zur Feier fünfzigjährigen Wirkens gewidmet|date=1874|pages=557–584|url=https://books.google.com/books?id=b4gEAAAAYAAJ}}</ref> The portion of the denominator <math> n\sin \theta </math> is called the [[numerical aperture]] (NA) and can reach about 1.4–1.6 in modern optics, hence the Abbe limit is <math>d=\frac{\lambda}{2.8}</math>.

Considering green light around 500&nbsp;nm and a NA of 1, the Abbe limit is roughly <math>d=\frac{\lambda}{2}=250 \text{ nm}</math> (0.25 μm), which is small compared to most biological cells (1 μm to 100 μm), but large compared to viruses (100&nbsp;nm), proteins (10&nbsp;nm) and less complex molecules (1&nbsp;nm). To increase the resolution, shorter wavelengths can be used such as UV and X-ray microscopes. These techniques offer better resolution but are expensive, suffer from lack of contrast in biological samples and may damage the sample.