Angular resolution

Angular resolution describes the ability of any image-forming device such as an 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.

The value that quantifies this property, θ, which is given by the Rayleigh criterion, is low for a system with a high resolution.

The Rayleigh criterion shows that the minimum angular spread that can be resolved by an image-forming system is limited by diffraction to the ratio of the wavelength of the waves to the aperture width.

Resolving power is the ability of an imaging device to separate (i.e., to see as distinct) points of an object that are located at a small angular distance or it is the power of an optical instrument to separate far away objects, that are close together, into individual images.

In scientific analysis, in general, the term "resolution" is 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.

On the other hand, diffraction comes from the wave nature of light and is determined by the finite aperture of the optical elements.

The interplay between diffraction and aberration can be characterised by the point spread function (PSF).

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,[1][2] 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.)

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.

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.

[3] Modern image processing techniques including deconvolution of the point spread function allow resolution of binaries with even less angular separation.

[Note 1] 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.

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 one arcsecond, but astronomical seeing and other atmospheric effects make attaining this very hard.

is the wavelength of light illuminating or emanating from (in the case of fluorescence microscopy) the sample.

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.

[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).

A series of images representing the magnification of M87* with an angular size of some microarcseconds , comparable to viewing a tennis ball on the Moon (magnification from top left corner counter−clockwise to the top right corner).
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.