Editing Angular resolution (section)

==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>