# Rayleigh Criterion

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. It is used in optics applied to light waves, in antenna theory applied to radio waves, and in acoustics applied to sound waves. The colloquial use of the term "resolution" sometimes causes confusion; when an optical system is said to have a high resolution or high angular resolution, it means that the perceived distance, or actual angular distance, between resolved neighboring objects is small. The value that quantifies this property, θ, which is given by the Rayleigh criterion, is low for a system with a high resolution. The closely related term spatial resolution refers to the precision of a measurement with respect to space, which is directly connected to angular resolution in imaging instruments. 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. For this reason, high resolution imaging systems such as astronomical telescopes, long distance telephoto camera lenses and radio telescopes have large apertures.

## Rayleigh Criterion

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. 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. As explained below, diffraction-limited resolution is defined by the Rayleigh criterion as the angular separation of two point sources when the maximum of each source lies in the first minimum of the diffraction pattern (Airy disk) of the other. 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.

The interplay between diffraction and aberration can be characterised by the point spread function (PSF). The narrower the aperture of a lens the more likely the PSF is dominated by diffraction. 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.) If the distance is greater, the two points are well resolved and if it is smaller, they are regarded as not resolved. Rayleigh defended this criterion on sources of equal strength.[2]

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. The result, θ = 4.56/D, with D in inches and θ in arcseconds, is slightly narrower than calculated with the Rayleigh criterion. A calculation using Airy discs as point spread function shows that at Dawes' limit there is a 5% dip between the two maxima, whereas at Rayleigh's criterion there is a 26.3% dip.[3] Modern image processing techniques including deconvolution of the point spread function allow resolution of binaries with even less angular separation.

To obtain a good image, point sources must be sufficiently far apart that their diffraction patterns do not overlap. To achieve this, the minimum distance between images must be such that the central maximum of the first image lies on the first minimum of the second and vice versa. Such an image is said to be just resolved. This is the famous Rayleigh criterion.

In most biology laboratories, resolution is presented when the use of the microscope is introduced. The ability of a lens to produce sharp images of two closely spaced point objects is called resolution. The smaller the distance x by which two objects can be separated and still be seen as distinct, the greater the resolution. The resolving power of a lens is defined as that distance x. An expression for resolving power is obtained from the Rayleigh criterion. In Figure 6(a) we have two point objects separated by a distance x. According to the Rayleigh criterion, resolution is possible when the minimum angular separation is

Just what is the limit? To answer that question, consider the diffraction pattern for a circular aperture, which has a central maximum that is wider and brighter than the maxima surrounding it (similar to a slit) (see Figure 2a). It can be shown that, for a circular aperture of diameter D, the first minimum in the diffraction pattern occurs at [latex]\theta=1.22\frac\lambdaD\\[/latex] (providing the aperture is large compared with the wavelength of light, which is the case for most optical instruments). The accepted criterion for determining the diffraction limit to resolution based on this angle was developed by Lord Rayleigh in the 19th century. The Rayleigh criterion for the diffraction limit to resolution states that two images are just resolvable when the center of the diffraction pattern of one is directly over the first minimum of the diffraction pattern of the other. See Figure 2b. The first minimum is at an angle of [latex]\theta=1.22\frac\lambdaD\\[/latex], so that two point objects are just resolvable if they are separated by the angle

The Rayleigh criterion stated in the equation [latex]\theta=1.22\frac\lambdaD\\[/latex] gives the smallest possible angle θ between point sources, or the best obtainable resolution. Once this angle is found, the distance between stars can be calculated, since we are given how far away they are.

In most biology laboratories, resolution is presented when the use of the microscope is introduced. The ability of a lens to produce sharp images of two closely spaced point objects is called resolution. The smaller the distance x by which two objects can be separated and still be seen as distinct, the greater the resolution. The resolving power of a lens is defined as that distance x. An expression for resolving power is obtained from the Rayleigh criterion. In Figure 6a we have two point objects separated by a distance x. According to the Rayleigh criterion, resolution is possible when the minimum angular separation is

Resolution can be defined as the minimum separation between two objects that results in a certain level of contrast between them. When two objects are brought together their PSFs combine additively and the total PSF of both objects is what is imaged by the microscope. When the objects are sufficiently far apart there is a dip in the intensity of the total PSF between the objects and they can be distinguished as separate entities and said to be resolved. The various microscopy lateral resolution limits, of which the Rayleigh criterion is but one, are essentially just different definitions of what constitutes a sufficient level of contrast between the objects for them to be resolved.

The Rayleigh criterion is named after English physicist John William Strutt, 3rd Baron Rayleigh (1842-1919) who investigated the image formation of telescopes and microscopes in the late 19th century. Rayleigh defined the resolution limit as the separation where the central maximum of the Airy pattern of one point emitter is directly overlapping with the first minimum of the Airy pattern of the other.1,2 In other words, the minimum resolvable separation between the points is the radius of the Airy disc which is given by:

Rayleigh chose his criterion based on the human visual system and to provide sufficient contrast for an observer to distinguish two separate objects in the image. The Rayleigh criterion is therefore not a fundamental physical law and instead a somewhat arbitrarily defined value. This was clearly stated by Rayleigh himself in 1879:2

We experimentally and numerically tested the separability of two independent equally luminous monochromatic and white light sources at the diffraction limit, using optical vortices (OV). The diffraction pattern of one of the two sources crosses a fork hologram on its center generating the Laguerre-Gaussian (LG) transform of an Airy disk. The second source, crossing the fork hologram in positions different from the optical center, generates nonsymmetric LG patterns. We formulated a criterion, based on the asymmetric intensity distribution of the superposed LG patterns so created, to resolve the two sources at angular distances much below the Rayleigh criterion. Analogous experiments in white light allow angular resolutions which are still one order of magnitude below the Rayleigh criterion. The use of OVs might offer new applications for stellar separation in future space experiments.

Super-resolution microscopy is a collective name for a number of techniques that achieve resolution below the conventional resolution limit, defined as the minimum distance that two point-source objects have to be in order to distinguish the two sources from each other. There are two closely related values for the diffraction limit, the Abbe and Rayleigh criterions. The difference between the two is based on the definition that both Abbe and Rayleigh used in their derivation for what is meant by two objects being resolvable from each other. In practical applications, this difference is small. The Abbe criterion is defined as: 041b061a72