Angular resolution

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Angular resolution describes the ability of any

The imaging system's resolution can be limited either by

Considering diffraction through a circular aperture, this translates into:

${\displaystyle \theta \approx 1.22{\frac {\lambda }{D}}\quad ({\text{considering that}}\,\sin \theta \approx \theta )}$

where θ is the angular resolution (

Bessel function of the first kind

${\displaystyle J_{1}(x)}$ divided by

The formal Rayleigh criterion is close to the

Using a small-angle approximation, the angular resolution may be converted into a spatial resolution, Δℓ, 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. For this case, the Rayleigh criterion reads:

${\displaystyle \Delta \ell \approx 1.22{\frac {f\lambda }{D}}}$.

This is the

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/#:

${\displaystyle \Delta \ell \approx 1.22{\frac {f\lambda }{D}}=1.22\lambda \cdot (f/\#)}$.

Since this is the radius of the Airy disk, the resolution is better estimated by the diameter, ${\displaystyle 2.44\lambda \cdot (f/\#)}$

Specific cases

Single telescope

Point-like sources separated by an

This formula, for light with a wavelength of about 562 nm, is also called the Dawes' limit.

Telescope array

The highest angular resolutions for telescopes can be achieved by arrays of telescopes called astronomical interferometers: These instruments can achieve angular resolutions of 0.001 arcsecond at optical wavelengths, and much higher resolutions at x-ray wavelengths. In order to perform aperture synthesis imaging, a large number of telescopes are required laid out in a 2-dimensional arrangement with a dimensional precision better than a fraction (0.25x) of the required image resolution.

The angular resolution R of an interferometer array can usually be approximated by

${\displaystyle R={\frac {\lambda }{B}}}$

where λ is the

The resolution R (here measured as a distance, not to be confused with the angular resolution of a previous subsection) depends on the angular aperture ${\displaystyle \alpha }$:^[5]

${\displaystyle R={\frac {1.22\lambda }{\mathrm {NA} _{\text{condenser}}+\mathrm {NA} _{\text{objective}}}}}$ where ${\displaystyle \mathrm {NA} =n\sin \theta }$.

Here NA is the numerical aperture, ${\displaystyle \theta }$ is half the included angle ${\displaystyle \alpha }$ of the lens, which depends on the diameter of the lens and its focal length, ${\displaystyle n}$ is the refractive index of the medium between the lens and the specimen, and ${\displaystyle \lambda }$ is the wavelength of light illuminating or emanating from (in the case of fluorescence microscopy) the sample.

It follows that the NAs of both the objective and the condenser should be as high as possible for maximum resolution. In the case that both NAs are the same, the equation may be reduced to:

${\displaystyle R={\frac {0.61\lambda }{\mathrm {NA} }}\approx {\frac {\lambda }{2\mathrm {NA} }}}$

The practical limit for ${\displaystyle \theta }$ is about 70°. In a dry objective or condenser, this gives a maximum NA of 0.95. In a high-resolution

(${\displaystyle \lambda \approx 400\,\mathrm {nm} }$),

${\displaystyle R={\frac {1.22\times 400\,{\mbox{nm}}}{1.45\ +\ 0.95}}=203\,{\mbox{nm}}}$

which is near 200 nm.

Oil immersion objectives can have practical difficulties due to their shallow depth of field and extremely short working distance, which calls for the use of very thin (0.17 mm) cover slips, or, in an inverted microscope, thin glass-bottomed Petri dishes.

However, resolution below this theoretical limit can be achieved using super-resolution microscopy. These include optical near-fields (Near-field scanning optical microscope) or a diffraction technique called 4Pi STED microscopy. Objects as small as 30 nm have been resolved with both techniques.^[6][7] In addition to this Photoactivated localization microscopy can resolve structures of that size, but is also able to give information in z-direction (3D).

List of telescopes and arrays by angular resolution

Name	Image	Angular resolution ( arc seconds )	Wavelength	Type	Site	Year
Global mm-VLBI Array (successor to the Coordinated Millimeter VLBI Array)		0.000012 (12 μas)	radio (at 1.3 cm)	very long baseline interferometry array of different radio telescopes	a range of locations on Earth and in space[8]	2002 -
PIONIER		0.001 (1 mas)	light (1-2 micrometre)[9]	largest optical array of 4 reflecting telescopes	Paranal Observatory, Antofagasta Region, Chile	2002/2010 -
Hubble Space Telescope		0.04	light (near 500 nm)[10]	space telescope	Earth orbit	1990 -
James Webb Space Telescope		0.1[11]	infrared (at 2000 nm)[12]	space telescope	Sun–Earth L2	2022 -

See also

• Angular diameter
• Beam diameter
• Dawes' limit
• Diffraction-limited system
• Ground sample distance
• Image resolution
• Optical resolution
• Sparrow's resolution limit
• Visual acuity

Notes

References