Angular resolution describes the resolving power of any image forming device such as an optical or radio telescope, a microscope, a camera, or an eye. An optical telescope is a Telescope which is used to gather and focus light mainly from the visible part of the Electromagnetic spectrum A radio telescope is a form of directional Radio antenna used in Radio astronomy and in tracking and collecting data from Satellites A microscope ( Greek: ( micron) = small + ( skopein) = to look or see is an instrument for viewing objects that are A camera is a device used to capture images either as still Photographs or as sequences of moving images ( Movies or Videos. Eyes are organs that detect Light, and send signals along the Optic nerve to the visual areas of the brain

## Definition of terms

Resolving power is the ability of the components of an imaging device to measure the angular separation of the points in an object. In Mathematics (in particular Geometry and Trigonometry) and all Natural sciences (including astronomy geophysics etc the angular distance 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. Optical resolution describes the ability of an imaging system to resolve detail in the object that is being imaged An object in Image processing is an identifiable portion of an image that can be interpreted as a single unit In scientific analysis the term "resolution" is generally used to describe the precision with which any instrument measures and records (in an image or spectrum) any variable in the specimen or sample under study. A variable (ˈvɛərɪəbl is an Attribute of a physical or an abstract System which may change its Value while it is under Observation.

## Explanation

The resolving power of a lens is ultimately limited by diffraction (see Point Spread Function, Airy disc). A lens is an optical device with perfect or approximate Axial symmetry which transmits and refracts Light, converging or diverging Diffraction is normally taken to refer to various phenomena which occur when a wave encounters an obstacle The point spread function ( PSF) describes the response of an imaging system to a Point source or point object The Airy disk (or Airy disc) is a phenomenon in Optics. Owing to the wave nature of light, light passing through an Aperture is diffracted 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 the Airy pattern, if the phase of the transmitted light is taken to be spherical over the exit aperture. Light, or visible light, is Electromagnetic radiation of a Wavelength that is visible to the Human eye (about 400–700 In physics interference is the addition ( superposition) of two or more Waves that result in a new wave pattern The Airy disk (or Airy disc) is a phenomenon in Optics. Owing to the wave nature of light, light passing through an Aperture is diffracted The phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0 The result is a blurring of the image. In Geometrical optics, a focus, also called an image point, is the point where Light rays originating from a point on the object converge. An empirical diffraction limit is given by the Rayleigh criterion invented by Lord Rayleigh:

 $\sin \theta = 1.220 \frac{\lambda}{D}$ whereθ is the angular resolution,λ is the wavelength of light,and D is the diameter of the lens' aperture. A central concept in Science and the Scientific method is that all Evidence must be empirical, or empirically based that is dependent on evidence John William Strutt 3rd Baron Rayleigh OM (12 November 1842 &ndash 30 June 1919 was an English Physicist who with William Ramsay, discovered In Physics wavelength is the distance between repeating units of a propagating Wave of a given Frequency. Geometry, a diameter of a Circle is any straight Line segment that passes through the center of the circle and whose Endpoints are on the

The factor 1. 220 is derived from a calculation of the position of the first dark ring surrounding the central Airy disc of the diffraction pattern. The Airy disk (or Airy disc) is a phenomenon in Optics. Owing to the wave nature of light, light passing through an Aperture is diffracted Diffraction is normally taken to refer to various phenomena which occur when a wave encounters an obstacle If one considers diffraction through a circular aperture, then the calculation involves a Bessel function -- 1. In Mathematics, Bessel functions, first defined by the Mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are Canonical 220 is approximately the first zero of the Bessel function of the first kind, of order one (i. e. J1), divided by π. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems This factor is used to approximate the ability of the human eye to distinguish two separate point sources depending on the overlap of their Airy discs: the minimum of one point source is located at the maximum of the other. Eyes are organs that detect Light, and send signals along the Optic nerve to the visual areas of the brain Modern telescopes and microscopes with video sensors may be slightly better than the human eye in their ability to discern overlap of Airy discs. A telescope is an instrument designed for the observation of remote objects and the collection of Electromagnetic radiation. A microscope ( Greek: ( micron) = small + ( skopein) = to look or see is an instrument for viewing objects that are A sensor is a device that measures a physical quantity and converts it into a signal which can be read by an observer or by an instrument 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 computerized image processing. Image processing is any form of Signal processing for which the input is an image such as photographs or frames of video the output of image processing can be either an image

For an ideal lens of focal length f, the Rayleigh criterion yields a minimum spatial resolution, Δl:

$\Delta l = 1.220 \frac{ f \lambda}{D}$. The focal length of an optical system is a measure of how strongly it converges (focuses or diverges (diffuses Light.

This is the size of smallest object that the lens can resolve, and also the radius of the smallest spot that a collimated beam of light can be focused to. Remote Authentication Dial In User Service ( RADIUS) is a networking protocol that provides centralized access authorization and accounting management for people or computers Collimated light is Light whose rays are nearly parallel and therefore will spread slowly as it propagates Light, or visible light, is Electromagnetic radiation of a Wavelength that is visible to the Human eye (about 400–700 The size is proportional to wavelength, λ, and thus, for example, blue light can be focused to a smaller spot than red light. Blue is a Colour, the Perception of which is evoked by Red is any of a number of similar Colors evoked by light consisting predominantly of the longest wavelengths of Light discernible by the human eye in the wavelength If the lens is focusing a beam of light with a finite extent (e. Light, or visible light, is Electromagnetic radiation of a Wavelength that is visible to the Human eye (about 400–700 g. , a laser beam), the value of D corresponds to the diameter of the light beam, not the lens. A laser is a device that emits Light ( Electromagnetic radiation) through a process called Stimulated emission. Geometry, a diameter of a Circle is any straight Line segment that passes through the center of the circle and whose Endpoints are on the 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. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called A single optical telescope may have an angular resolution less than one arcsecond, but astronomical seeing and other atmospheric effects make attaining this very hard. A minute of arc, arcminute, or MOA is a unit of angular measurement, equal to one sixtieth (1/60 of one degree. Astronomical seeing refers to the blurring and twinkling of astronomical objects such as stars caused by Turbulence in the Earth's atmosphere

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. In Physics wavelength is the distance between repeating units of a propagating Wave of a given Frequency. An objective in Optics is the lens or Mirror in a Microscope, Telescope, camera or other optical instrument

Resulting R is in radians. The radian is a unit of plane Angle, equal to 180/ π degrees, or about 57 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. Dawes' limit is a formula to express the maximum Resolving power of a Microscope or Telescope.

## Telescope array case

The highest angular resolutions can be achieved by arrays of telescopes called astronomical interferometers: these instruments can achieve angular resolutions of 0. An astronomical interferometer is an array of telescopes or mirror segments acting together to probe structures with higher resolution 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 the baseline. In Physics wavelength is the distance between repeating units of a propagating Wave of a given Frequency.

The resulting R is in radians. The radian is a unit of plane Angle, equal to 180/ π degrees, or about 57 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 $\times$ 120 m.

## Microscope case

The resolution R depends on the angular aperture α:

$R=\frac{1.22\lambda}{2\times N.A.}=\frac{1.22\lambda}{2n\sin\theta}$. The angular aperture of a lens is the apparent Angle of the lens Aperture as seen from the focal point: a = 2 \arctan [1]

Here θ is the collecting angle of the lens, which depends on the width of objective lens and its focal distance from the specimen. An objective in Optics is the lens or Mirror in a Microscope, Telescope, camera or other optical instrument n is the refractive index of the medium in which the lens operates. The refractive index (or index of Refraction) of a medium is a measure for how much the speed of light (or other waves such as sound waves is reduced inside the medium λ is the wavelength of light illuminating or emanating from (in the case of fluorescence microscopy) the sample. The quantity $n\times\sin\theta$ is also known as the numerical aperture. In Optics, the numerical aperture ( NA) of an optical system is a Dimensionless number that characterizes the range of angles over which the system can accept

Due to the limitations of the values θ, λ, and n, the resolution limit of a light microscope using visible light is about 200 nm. A nanometre ( American spelling: nanometer, symbol nm) ( Greek: νάνος nanos dwarf; μετρώ metrό count) is a This is because: α for the best lens is about 70° (sin α = 0. 94), the shortest wavelength of visible light is blue (λ = 450nm), and the typical high resolution lenses are oil immersion lenses (n = 1. In Light microscopy, oil immersion is a technique used to increase the resolution of a Microscope. 56):

$R=\frac{0.61 \times 450\,\mbox{nm}}{1.56 \times 0.94} = 187\,\mbox{nm}$

## References

1. ^ Nikon MicroscopyU: Concepts and Formulas in Microscopy: Resolution