Angular resolution

Airy diffraction patterns generated by light from two point sources 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.
Log-log plot of aperture diameter vs angular resolution at the diffraction limit for various light wavelengths compared with various astronomical instruments. For example, the blue star shows that the Hubble Space Telescope is almost diffraction-limited in the visible spectrum at 0.1 arcsecs, whereas the red circle shows that the human eye should have a resolving power of 20 arcsecs in theory, though normally only 60 arcsecs.

Optical or radio telescope, a microscope, a camera, or an eye, to distinguish small details of an object, thereby making it a major determinant of image resolution.

- Angular resolution
Airy diffraction patterns generated by light from two point sources 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.

213 related topics

Relevance

Various examples of physical phenomena

Spatial resolution

In physics and geosciences, the term spatial resolution refers to the linear spacing of a measurement, or the physical dimension that represents a pixel of the image.

In physics and geosciences, the term spatial resolution refers to the linear spacing of a measurement, or the physical dimension that represents a pixel of the image.

Various examples of physical phenomena

While in some instruments, like cameras and telescopes, spatial resolution is directly connected to angular resolution, other instruments, like synthetic aperture radar or a network of weather stations, produce data whose spatial sampling layout is more related to the Earth's surface, such as in remote sensing and satellite imagery.

John William Strutt, 3rd Baron Rayleigh

John William Strutt, 3rd Baron Rayleigh

English mathematician who made extensive contributions to science.

English mathematician who made extensive contributions to science.

John William Strutt, 3rd Baron Rayleigh
Caricature of Lord Rayleigh in the London magazine Vanity Fair, 1899
Theory of sound, 1894

In optics, Rayleigh proposed a well-known criterion for angular resolution.

The 64-meter radio telescope at Parkes Observatory as seen in 1969, when it was used to receive live televised video from Apollo 11

Radio telescope

Specialized antenna and radio receiver used to detect radio waves from astronomical radio sources in the sky.

Specialized antenna and radio receiver used to detect radio waves from astronomical radio sources in the sky.

The 64-meter radio telescope at Parkes Observatory as seen in 1969, when it was used to receive live televised video from Apollo 11
Antenna of UTR-2 low frequency radio telescope, Kharkiv region, Ukraine. Consists of an array of 2040 cage dipole elements.
Ooty radio telescope, a 326.5 MHz dipole array in Ooty, India
Plot of Earth's atmospheric transmittance (or opacity) to various wavelengths of electromagnetic radiation.
The Very Large Array in Socorro, New Mexico, an interferometric array formed of 27 parabolic dish telescopes.
Atacama Large Millimeter Array in the Atacama desert consisting of 66 12-metre (39 ft), and 7-metre (23 ft) diameter radio telescopes designed to work at sub-millimeter wavelengths
alt=Five hundred meter Aperture Spherical Telescope under construction|The 500 meter Five hundred meter Aperture Spherical Telescope (FAST), under construction, China (2016)
alt=Green Bank Telescope|The 100 meter Green Bank Telescope, Green Bank, West Virginia, US, the largest fully steerable radio telescope dish (2002)
alt=Effelsberg 100-m Radio Telescope|The 100 meter Effelsberg, in Bad Münstereifel, Germany (1971)
alt=Lovell Telescope|The 76 meter Lovell, Jodrell Bank Observatory, England (1957)
alt=DSS 14 "Mars" antenna at Goldstone Deep Space Communications Complex|The 70 meter DSS 14 "Mars" antenna at Goldstone Deep Space Communications Complex, Mojave Desert, California, US (1958)
alt=Yevpatoria RT-70 radio telescope|The 70 meter Yevpatoria RT-70, Crimea, first of three RT-70 in the former Soviet Union, (1978)
The 70 meter Galenki RT-70, Galenki, Russia, second of three RT-70 in the former Soviet Union, (1984)

The angular resolution of a dish antenna is determined by the ratio of the diameter of the dish to the wavelength of the radio waves being observed.

Schematic diagram illustrating how optical wavefronts from a distant star may be perturbed by a layer of turbulent mixing in the atmosphere. The vertical scale of the wavefronts plotted is highly exaggerated.

Astronomical seeing

Astronomical object due to turbulent airflows in the atmosphere of Earth that may become visible as blurring, twinkling or variable distortion.

Astronomical object due to turbulent airflows in the atmosphere of Earth that may become visible as blurring, twinkling or variable distortion.

Schematic diagram illustrating how optical wavefronts from a distant star may be perturbed by a layer of turbulent mixing in the atmosphere. The vertical scale of the wavefronts plotted is highly exaggerated.
Simulated negative image showing what a single (point-like) star would look like through a ground-based telescope with a diameter of 2r0. The blurred look of the image is because of diffraction, which causes the appearance of the star to be an Airy pattern with a central disk surrounded by hints of faint rings. The atmosphere would make the image move around very rapidly, so that in a long-exposure photograph it would appear more blurred.
Simulated negative image showing what a single (point-like) star would look like through a ground-based telescope with a diameter of 7r0, on the same angular scale as the 2r0 image above. The atmosphere makes the image break up into several blobs (speckles). The speckles move around very rapidly, so that in a long-exposure photograph the star would appear as a single blurred blob.
Simulated negative image showing what a single (point-like) star would look like through a ground-based telescope with a diameter of 20r0. The atmosphere makes the image break up into several blobs (speckles). The speckles move around very rapidly, so that in a long-exposure photograph the star would appear as a single blurred blob.
Astronomical observatories are generally situated on mountaintops, as the air at ground level is usually more convective. A light wind bringing stable air from high above the clouds and ocean generally provides the best seeing conditions (telescope shown: NOT).
An animated image of the Moon's surface showing the effects of Earth's atmosphere on the view
Astronomers can make use of an artificial star by shining a powerful laser to correct for the blurring caused by the atmosphere.
This amateur lucky imaging stack using the best of 1800 frames of Jupiter captured using a relatively small telescope approaches the theoretical maximum resolution for the telescope, rather than being limited by seeing.
Slow motion movie of the image seen at a telescope when looking at a star at high magnification (negative images). The telescope used had a diameter of about 7r{{sub|0}} (see definition of r{{sub|0}} below, and example simulated image through a 7r{{sub|0}} telescope). The star breaks up into multiple blobs (speckles) -- entirely an atmospheric effect. Some telescope vibration is also noticeable.

Seeing is a major limitation to the angular resolution in astronomical observations with telescopes that would otherwise be limited through diffraction by the size of the telescope aperture.

A diffraction pattern of a red laser beam projected onto a plate after passing through a small circular aperture in another plate

Diffraction

Obstacle or opening.

Obstacle or opening.

A diffraction pattern of a red laser beam projected onto a plate after passing through a small circular aperture in another plate
Infinitely many points (three shown) along length d project phase contributions from the wavefront, producing a continuously varying intensity θ on the registering plate.
Thomas Young's sketch of two-slit diffraction for water waves, which he presented to the Royal Society in 1803.
Photograph of single-slit diffraction in a circular ripple tank
Circular waves generated by diffraction from the narrow entrance of a flooded coastal quarry
A solar glory on steam from hot springs. A glory is an optical phenomenon produced by light backscattered (a combination of diffraction, reflection and refraction) towards its source by a cloud of uniformly sized water droplets.
2D Single-slit diffraction with width changing animation
Numerical approximation of diffraction pattern from a slit of width four wavelengths with an incident plane wave. The main central beam, nulls, and phase reversals are apparent.
Graph and image of single-slit diffraction.
2-slit (top) and 5-slit diffraction of red laser light
Diffraction of a red laser using a diffraction grating.
A diffraction pattern of a 633 nm laser through a grid of 150 slits
A computer-generated image of an Airy disk.
Computer generated light diffraction pattern from a circular aperture of diameter 0.5 micrometre at a wavelength of 0.6 micrometre (red-light) at distances of 0.1 cm – 1 cm in steps of 0.1 cm. One can see the image moving from the Fresnel region into the Fraunhofer region where the
Airy pattern is seen.
The Airy disk around each of the stars from the 2.56 m telescope aperture can be seen in this lucky image of the binary star zeta Boötis.
The upper half of this image shows a diffraction pattern of He-Ne laser beam on an elliptic aperture. The lower half is its 2D Fourier transform approximately reconstructing the shape of the aperture.
Following Bragg's law, each dot (or reflection) in this diffraction pattern forms from the constructive interference of X-rays passing through a crystal. The data can be used to determine the crystal's atomic structure.
Computer generated intensity pattern formed on a screen by diffraction from a square aperture.
Generation of an interference pattern from two-slit diffraction.
Computational model of an interference pattern from two-slit diffraction.
Optical diffraction pattern ( laser), (analogous to X-ray crystallography)
Colors seen in a spider web are partially due to diffraction, according to some analyses.<ref>{{cite web|url = http://www.itp.uni-hannover.de/%7Ezawischa/ITP/spiderweb.html|title = Optical effects on spider webs|author = Dietrich Zawischa|access-date = 2007-09-21}}</ref>
Diffraction on a sharp metallic edge
Diffraction on a soft aperture, with a gradient of conductivity over the image width

The Rayleigh criterion specifies that two point sources are considered "resolved" if the separation of the two images is at least the radius of the Airy disk, i.e. if the first minimum of one coincides with the maximum of the other.

Several objective lenses on a microscope.

Objective (optics)

Optical element that gathers light from the object being observed and focuses the light rays to produce a real image.

Optical element that gathers light from the object being observed and focuses the light rays to produce a real image.

Several objective lenses on a microscope.
Objective lenses of binoculars
Two Leica oil immersion microscope objective lenses; left 100×, right 40×.
Camera photographic objective, focal length 50 mm, aperture 1:1.4
The segmented hexagonal objective mirror of the Keck 2 Telescope

A telescope's light-gathering power and angular resolution are both directly related to the diameter (or "aperture") of its objective lens or mirror.

The Large Binocular Telescope uses two curved mirrors to gather light

Optical telescope

Telescope that gathers and focuses light mainly from the visible part of the electromagnetic spectrum, to create a magnified image for direct visual inspection, to make a photograph, or to collect data through electronic image sensors.

Telescope that gathers and focuses light mainly from the visible part of the electromagnetic spectrum, to create a magnified image for direct visual inspection, to make a photograph, or to collect data through electronic image sensors.

The Large Binocular Telescope uses two curved mirrors to gather light
Schematic of a Keplerian refracting telescope. The arrow at (4) is a (notional) representation of the original image; the arrow at (5) is the inverted image at the focal plane; the arrow at (6) is the virtual image that forms in the viewer's visual sphere. The red rays produce the midpoint of the arrow; two other sets of rays (each black) produce its head and tail.
Eight-inch refracting telescope at Chabot Space and Science Center
The Keck II telescope gathers light by using 36 segmented hexagonal mirrors to create a 10 m (33 ft) aperture primary mirror
These eyes represent a scaled figure of the human eye where 15 px = 1 mm, they have a pupil diameter of 7 mm. Figure A has an exit pupil diameter of 14 mm, which for astronomy purposes results in a 75% loss of light. Figure B has an exit pupil of 6.4 mm, which allows the full 100% of observable light to be perceived by the observer.
Two of the four Unit Telescopes that make up the ESO's VLT, on a remote mountaintop, 2600 metres above sea level in the Chilean Atacama Desert.
Comparison of nominal sizes of primary mirrors of some notable optical telescopes
Harlan J. Smith Telescope reflecting telescope at McDonald Observatory, Texas

It is analogous to angular resolution, but differs in definition: instead of separation ability between point-light sources it refers to the physical area that can be resolved.

The wavelength of a sine wave, λ, can be measured between any two points with the same phase, such as between crests (on top), or troughs (on bottom), or corresponding zero crossings as shown.

Wavelength

Spatial period of a periodic wave—the distance over which the wave's shape repeats.

Spatial period of a periodic wave—the distance over which the wave's shape repeats.

The wavelength of a sine wave, λ, can be measured between any two points with the same phase, such as between crests (on top), or troughs (on bottom), or corresponding zero crossings as shown.
Sinusoidal standing waves in a box that constrains the end points to be nodes will have an integer number of half wavelengths fitting in the box.
A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue)
Wavelength is decreased in a medium with slower propagation.
Refraction: upon entering a medium where its speed is lower, the wave changes direction.
Separation of colors by a prism (click for animation)
Various local wavelengths on a crest-to-crest basis in an ocean wave approaching shore
A sinusoidal wave travelling in a nonuniform medium, with loss
A wave on a line of atoms can be interpreted according to a variety of wavelengths.
Near-periodic waves over shallow water
Wavelength of a periodic but non-sinusoidal waveform.
A propagating wave packet
Pattern of light intensity on a screen for light passing through two slits. The labels on the right refer to the difference of the path lengths from the two slits, which are idealized here as point sources.
Diffraction pattern of a double slit has a single-slit envelope.
Relationship between wavelength, angular wavelength, and other wave properties.

Diffraction is the fundamental limitation on the resolving power of optical instruments, such as telescopes (including radiotelescopes) and microscopes.

ESO’s VLT interferometer took the first detailed image of a disc around a young star.

Astronomical interferometer

Set of separate telescopes, mirror segments, or radio telescope antennas that work together as a single telescope to provide higher resolution images of astronomical objects such as stars, nebulas and galaxies by means of interferometry.

Set of separate telescopes, mirror segments, or radio telescope antennas that work together as a single telescope to provide higher resolution images of astronomical objects such as stars, nebulas and galaxies by means of interferometry.

ESO’s VLT interferometer took the first detailed image of a disc around a young star.
A 20-foot Michelson interferometer mounted on the frame of the 100-inch Hooker Telescope, 1920.
Aerial view of the ESO/NAOJ/NRAO ALMA construction site.
The Navy Precision Optical Interferometer (NPOI), a 437 ma baselined optical/near-infrared, 6-beam Michelson Interferometer at 2163 m elevation on Anderson Mesa in Northern Arizona, USA. Four additional 1.8-meter telescopes are being installed starting from 2013.
Light collected by three ESO VLT auxiliary telescopes, and combined using the technique of interferometry.
This image shows one of a series of sophisticated optical and mechanical systems called star separators for the Very Large Telescope Interferometer (VLTI).
Two of the Atacama Large Millimeter/submillimeter array (ALMA) 12-metre antennas gaze at the sky at the observatory's Array Operations Site (AOS), high on the Chajnantor plateau at an altitude of 5000 metres in the Chilean Andes.

The advantage of this technique is that it can theoretically produce images with the angular resolution of a huge telescope with an aperture equal to the separation between the component telescopes.

Most aperture synthesis interferometers use the rotation of the Earth to increase the number of baseline orientations included in an observation. In this example with the Earth represented as a grey sphere, the baseline between telescope A and telescope B changes angle with time as viewed from the radio source as the Earth rotates. Taking data at different times thus provides measurements with different telescope separations.

Aperture synthesis

Most aperture synthesis interferometers use the rotation of the Earth to increase the number of baseline orientations included in an observation. In this example with the Earth represented as a grey sphere, the baseline between telescope A and telescope B changes angle with time as viewed from the radio source as the Earth rotates. Taking data at different times thus provides measurements with different telescope separations.

Aperture synthesis or synthesis imaging is a type of interferometry that mixes signals from a collection of telescopes to produce images having the same angular resolution as an instrument the size of the entire collection.