Editing Diffraction-limited system (section)

==Calculation of diffraction limit==

===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.

===Digital photography===

In a digital camera, diffraction effects interact with the effects of the regular pixel grid. The combined effect of the different parts of an optical system is determined by the [[convolution]] of the [[point spread function]]s (PSF). The point spread function of a diffraction limited circular-aperture lens is simply the [[Airy disk]]. The point spread function of the camera, otherwise called the instrument response function (IRF) can be approximated by a rectangle function, with a width equivalent to the pixel pitch. A more complete derivation of the modulation transfer function (derived from the PSF) of image sensors is given by Fliegel.<ref>{{cite journal|last1=Fliegel|first1=Karel|title=Modeling and Measurement of Image Sensor Characteristics|journal=Radioengineering|date=December 2004|volume=13|issue=4|url=http://www.radioeng.cz/fulltexts/2004/04_04_27_34.pdf}}</ref> Whatever the exact instrument response function, it is largely independent of the f-number of the lens. Thus at different f-numbers a camera may operate in three different regimes, as follows:

# In the case where the spread of the IRF is small with respect to the spread of the diffraction PSF, in which case the system may be said to be essentially diffraction limited (so long as the lens itself is diffraction limited).
# In the case where the spread of the diffraction PSF is small with respect to the IRF, in which case the system is instrument limited.
# In the case where the spread of the PSF and IRF are similar, in which case both impact the available resolution of the system.

The spread of the diffraction-limited PSF is approximated by the diameter of the first null of the [[Airy disk]],

:<math> d/2 = 1.22 \lambda N,\, </math><ref>{{cite book | last=Goodman | first=Joseph W. | chapter= 4.4.2 Example of Fraunhofer Diffraction Patterns for Circular Aperture | title=Introduction to Fourier Optics | publisher=Roberts and Company Publishers | publication-place=Englewood, Colorado | date=2005 | isbn=0-9747077-2-4}}</ref>

where <math>\lambda</math> is the wavelength of the light and <math>N</math> is the [[f-number]] of the imaging optics, i.e., <math> 2 NA \rightarrow (2.44N)^{-1} </math> in the Abbe diffraction limit formula. For instance, for an f/8 lens (<math>N=8</math> and <math>NA\approx2.5%</math> ) and for green light (<math>\lambda_g=</math> 0.5&nbsp;μm wavelength) light, the focusing spot diameter will be d = 9.76&nbsp;μm or 19.5<math>\lambda_g</math>. This is similar to the pixel size for the majority of commercially available 'full frame' (43mm sensor diagonal) cameras and so these will operate in regime 3 for f-numbers around 8 (few lenses are close to diffraction limited at f-numbers smaller than 8). Cameras with smaller sensors will tend to have smaller pixels, but their lenses will be designed for use at smaller f-numbers and it is likely that they will also operate in regime 3 for those f-numbers for which their lenses are diffraction limited.