- Angular resolution
**Angular resolution**describes the resolving power of any "image forming device" such as an optical orradio telescope , amicroscope , acamera , or aneye .**Definition of terms****Resolving power**is the ability of the components of an imaging device to measure theangular separation of the points in an object. The term**resolution**or**minimum resolvable distance**is the minimum distance between distinguishable objects in an image, although the term is loosely used by many users of microscopes and telescopes to describe resolving power. In scientific analysis the term "resolution" is generally used to describe theprecision with which any instrument measures and records (in an image or spectrum) anyvariable in the specimen or sample under study.**Explanation**The resolving power of a lens is ultimately limited by

diffraction (seePoint Spread Function ,Airy disc ). The lens'aperture is analogous to a two-dimensional version of the single-slit experiment.Light passing through the lens interferes with itself creating a ring-shaped diffraction pattern, known as theAiry pattern , if the phase of the transmitted light is taken to be spherical over the exit aperture. The result is a blurring of the image. Anempirical diffraction limit is given by the**Rayleigh criterion**invented byLord Rayleigh ::

The factor 1.220 is derived from a calculation of the position of the first dark ring surrounding the central

Airy disc of thediffraction pattern. If one considers diffraction through a circular aperture, then the calculation involves aBessel function -- 1.220 is approximately the first zero of the Bessel function of the first kind, of order one (i.e. $J\_\{1\}$), divided by π. This factor is used to approximate the ability of the humaneye to distinguish two separate point sources depending on theoverlap of their Airy discs: the minimum of one point source is located at the maximum of the other. Moderntelescope s andmicroscope s with videosensors may be slightly better than the human eye in their ability to discern overlap of Airy discs. Thus it is worth bearing in mind that the Rayleigh criterion is an empirical estimate of resolution based on the assumption of a human observer, and may slightly underestimate the resolving power of a particular optical train. For specialized imaging, foreknowledge of some characteristics of the image can also improve on technical resolution limits through computerizedimage processing .For an ideal lens of

focal length "f", the Rayleigh criterion yields a minimum**spatial resolution**, Δ"l"::$Delta\; l\; =\; 1.220\; frac\{\; f\; lambda\}\{D\}$.

This is the size of smallest object that the lens can resolve, and also the

radius of the smallest spot that acollimated beam oflight can be focused to. The size is proportional to wavelength, "λ", and thus, for example,blue light can be focused to a smaller spot thanred light. If the lens is focusing a beam oflight with a finite extent (e.g., alaser beam), the value of "D" corresponds to thediameter of the light beam, not the lens. 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 to a smaller spot than a narrow one.**Single telescope case**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 onearcsecond , butastronomical seeing and other atmospheric effects make attaining this very hard.The angular resolution "R" of a telescope can usually be approximated by:$R\; =\; frac\; \{lambda\}\{D\}$where:"λ" is the

wavelength of the observed radiation:and "D" is the diameter of the telescope's objective.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 the case of yellow light with a wavelength of 580 nm, for a resolution of 0.1 arc second, we need D = 1.2 m.

This formula, for light with a wavelength of ca 562 nm, is also called the

Dawes' limit .**Telescope array case**The highest angular resolutions can be achieved by arrays of telescopes called

astronomical interferometer s: these instruments can achieve angular resolutions of 0.001 arcsecond at optical wavelengths, and much higher resolutions at radio wavelengths. In order to perform aperture synthesis imaging, a large number of telescopes are required laid out in a 2-dimensional arrangement.The angular resolution "R" of an interferometer array can usually be approximated by:$R\; =\; frac\; \{lambda\}\{B\}$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 thebaseline .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 nm, for a resolution of 1 milli-arcsecond, we need telescopes laid out in an array which is 120 m × 120 m.

**Microscope case**The resolution "R" depends on the

angular aperture α::$R=frac\{1.22lambda\}\{2\; imes\; N.A.\}=frac\{1.22lambda\}\{2nsin\; heta\}$. [

*[*]*http://www.microscopyu.com/articles/formulas/formulasresolution.html Nikon MicroscopyU: Concepts and Formulas in Microscopy: Resolution*]Here "$heta$" is the "collecting angle" of the lens, which depends on the width of

objective lens and its focal distance from the specimen. "n" is the "refractive index " of the medium in which the lens operates. "λ" is the wavelength of light illuminating or emanating from (in the case of fluorescence microscopy) the sample. The quantity n × sin $heta$ is also known as thenumerical aperture or N.A.Due to the limitations of the values "$heta$", "λ", and "n", the resolution limit of a light microscope using

visible light is about 200 nm. This is because: "α" for the best lens is about 70° (sin "α" = 0.94), the shortest wavelength of visible light is blue ("λ" = 450 nm), and the typical high resolution lenses are oil immersion lenses ("n" = 1.56)::$R=frac\{1.22\; imes\; 450,mbox\{nm\{2\; imes\; 1.56\; imes\; 0.94\}\; =\; 187,mbox\{nm\}$

**References****See also***

Angular diameter

*Dawes limit

*Visual acuity **External links*** [

*http://www.microscopyu.com/articles/formulas/formulasresolution.html "Concepts and Formulas in Microscopy: Resolution"*] by Michael W. Davidson, "Nikon MicroscopyU" (website).

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