Airy disk

The disk and rings phenomenon had been known prior to Airy; John Herschel described the appearance of a bright star seen through a telescope under high magnification for an 1828 article on light for the Encyclopedia Metropolitana: ...the star is then seen (in favourable circumstances of tranquil atmosphere, uniform temperature, etc.)

as a perfectly round, well-defined planetary disc, surrounded by two, three, or more alternately dark and bright rings, which, if examined attentively, are seen to be slightly coloured at their borders.

They succeed each other nearly at equal intervals round the central disc....[1]Airy wrote the first full theoretical treatment explaining the phenomenon (his 1835 "On the Diffraction of an Object-glass with Circular Aperture").

Due to diffraction, the smallest point to which a lens or mirror can focus a beam of light is the size of the Airy disk.

Far from the aperture, the angle at which the first minimum occurs, measured from the direction of incoming light, is given by the approximate formula:

The Rayleigh criterion for barely resolving two objects that are point sources of light, such as stars seen through a telescope, is that the center of the Airy disk for the first object occurs at the first minimum of the Airy disk of the second.

However, while the angle at which the first minimum occurs (which is sometimes described as the radius of the Airy disk) depends only on wavelength and aperture size, the appearance of the diffraction pattern will vary with the intensity (brightness) of the light source.

In astronomy, the outer rings are frequently not apparent even in a highly magnified image of a star.

It may be that none of the rings are apparent, in which case the star image appears as a disk (central maximum only) rather than as a full diffraction pattern.

[7] In reality, the angle of first minimum is a limiting value for the size of the Airy disk, and not a definite radius.

The fastest f-number for the human eye is about 2.1,[8] corresponding to a diffraction-limited point spread function with approximately 1 μm diameter.

However, at this f-number, spherical aberration limits visual acuity, while a 3 mm pupil diameter (f/5.7) approximates the resolution achieved by the human eye.

[9] The maximum density of cones in the human fovea is approximately 170,000 per square millimeter,[10] which implies that the cone spacing in the human eye is about 2.5 μm, approximately the diameter of the point spread function at f/5.

However the exact Airy pattern does appear at a finite distance if a lens is placed at the aperture.

Then the Airy pattern will be perfectly focussed at the distance given by the lens's focal length (assuming collimated light incident on the aperture) given by the above equations.

of the first dark ring on the focal plane is solely given by the numerical aperture A (closely related to the f-number) by

Viewing the aperture of radius d/2 and lens as a camera (see diagram above) projecting an image onto a focal plane at distance f, the numerical aperture A is related to the commonly-cited f-number N= f/d (ratio of the focal length to the lens diameter) according to

This shows that the best possible image resolution of a camera is limited by the numerical aperture (and thus f-number) of its lens due to diffraction.

above can be integrated to give the total power contained in the diffraction pattern within a circle of given size:

[13] Classical treatments of the Airy disk and diffraction pattern assume that the incident light is a plane wave that consists of coherent (in phase) photons of the same wavelength that interfere with each other.

The famous double slit experiment showed that diffraction patterns could arise even when the coherent photons were so spread out in time that they could not interfere with each other.

This led to the quantum mechanical picture that each photon effectively takes all possible paths from a source to a detector.

The detection probability is the square of the modulus of the sum of the complex amplitudes at the detector.

According to these principles the Airy disk and diffraction pattern can be computed numerically by using Feynman photon path integrals to determine the detection probability at different points in the focal plane of a parabolic mirror.

As a result, the root mean square (RMS) spotsize is undefined (i.e. infinite).

An alternative measure of the spot size is to ignore the relatively small outer rings of the Airy pattern and to approximate the central lobe with a Gaussian profile, such that

If, on the other hand, we wish to enforce that the Gaussian profile has the same volume as does the Airy pattern, then this becomes In optical aberration theory, it is common to describe an imaging system as diffraction-limited if the Airy disk radius is larger than the RMS spotsize determined from geometric ray tracing (see Optical lens design).

Similar equations can also be derived for the obscured Airy diffraction pattern[16][17] which is the diffraction pattern from an annular aperture or beam, i.e. a uniform circular aperture (beam) obscured by a circular block at the center.

This becomes more problematic with short focal length telescopes which require larger secondary mirrors.

is This is the special case of the Fourier integral of the elliptical cross section with half axes

A computer-generated image of an Airy disk. The grayscale intensities have been adjusted to enhance the brightness of the outer rings of the Airy pattern.
A computer-generated Airy disk from diffracted white light ( D65 spectrum ). Note that the red component is diffracted more than the blue, so that the center appears slightly bluish.
A real Airy disk created by passing a red laser beam through a 90- micrometre pinhole aperture with 27 orders of diffraction
Airy disk captured by 2000 mm camera lens at f/25 aperture. Image size: 1×1 mm.
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 20/20 vision resolves to only 60 arcsecs (1 arcminute)
Longitudinal sections through a focused beam with (top) negative, (center) zero, and (bottom) positive spherical aberration. The lens is to the left.
Diffraction from a circular aperture. The Airy pattern is observable when (i.e. in the far field)
Diffraction from an aperture with a lens. The far field image will (only) be formed at the screen one focal length away, where R=f (f=focal length). The observation angle stays the same as in the lensless case.
The Airy Pattern on the interval ka sin θ = [−10, 10]
The encircled power graphed next to the intensity.
A radial cross-section through the Airy pattern (solid curve) and its Gaussian profile approximation (dashed curve). The abscissa is given in units of the wavelength times the f-number of the optical system.