# Angular resolution

### Angular resolution

Angular resolution or spatial 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.

## Contents

• Definition of terms 1
• Explanation 2
• Specific cases 3
• Single telescope 3.1
• Telescope array 3.2
• Microscope 3.3
• Notes 4
• References 6
• External links 7

## Definition of terms

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. 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.

## Explanation

Airy diffraction patterns generated by light from two points 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.

The imaging system's resolution can be limited either by aberration or by diffraction causing blurring of the image. These two phenomena have different origins and are unrelated. Aberrations can be explained by geometrical optics and can in principle be solved by increasing the optical quality — and subsequently the cost — of the system. On the other hand, diffraction comes from the wave nature of light and is determined by the finite aperture of the optical elements. The lens' circular aperture is analogous to a two-dimensional version of the single-slit experiment. Light passing through the lens interferes with itself creating a ring-shape diffraction pattern, known as the Airy pattern, if the wavefront of the transmitted light is taken to be spherical or plane over the exit aperture.

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 invented by Lord Rayleigh:

Two point sources are regarded as just resolved when the principal diffraction maximum of one image coincides with the first minimum of the other.[1] [2] 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 criteria on sources of equal strength.[2]

If one considers diffraction through a circular aperture, this translates into:

 \theta = 1.220 \frac{\lambda}{D} where θ is the angular resolution (radians), λ is the wavelength of light, and D is the diameter of the lens' aperture.

The factor 1.220 is derived from a calculation of the position of the first dark circular ring surrounding the central Airy disc of the diffraction pattern. The calculation involves a Bessel function—1.220 is approximately the first zero of the Bessel function of the first kind, of order one (i.e., J_{1}), divided by π.

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 even narrower binaries.

The angular resolution may be converted into a spatial resolution, Δ, 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:

\Delta \ell = 1.220 \frac{ f \lambda}{D}.

This is the size, in the imaging plane, of smallest object that the lens can resolve, and also the radius of the smallest spot to which a collimated beam of light can be focused.[4] The size is proportional to wavelength, λ, and thus, for example, blue light can be focused to a smaller spot than red light. If the lens is focusing a beam of light with a finite extent (e.g., a laser beam), the value of D corresponds to the diameter of the light beam, not the lens. 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. This result is related to the Fourier properties of a lens.

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

• "Concepts and Formulas in Microscopy: Resolution" by Michael W. Davidson, Nikon MicroscopyU (website).
• Melles Griot Technical Guide: [3].
• Melles Griot Technical Guide: [4].