Airy disk

Airy disk arising from the diffraction of a light beam at an aperture. Is the iris circular, one observes a central maximum surrounded by rings of decreasing light radiation intensity. Other designations are diffraction ring or Airy disk. Non -circular aperture also produce diffraction patterns that can be clearly distinguished from an Airy disk (spikes). Mathematics describes the diffraction of light by the diffraction integral.

Phenomenology

Even a perfect accordance with the laws of geometrical optics instrument without aberrations, can a given object as a point of light not map exactly to a point, but is caused by the diffraction of light at the aperture in the image plane a blur spot. The shape of the spot depends on the reciprocal on the shape of the aperture, in particular its size is inversely proportional to the size of the aperture. For a circular aperture, given approximately by the round version of a lens, and the stain is rotationally symmetrical, with a central maximum and weak, concentric rings. Since the size of this pattern also depends on the wavelength, in white light, the diffraction rings are barely visible. The central diffraction spot is after the English astronomer George Biddell Airy also called Airy disk.

Although an attempt is adjacent points of an object and distinguish increased over the image distance, the magnification so grows the distance between the corresponding diffraction patterns, but also the diffraction patterns themselves are larger in the same proportion. This is called diffraction limit of the angular resolution capability. Easy to calculate diffraction patterns are for infinite image distance. This corresponds to the pinhole camera and the Fresnel approximation of the diffraction integral.

Because of the small aperture with the expectant diffraction disk on one side and with the aperture increasing spherical aberration on the other hand, the largest sharpness results in an optical imaging at the critical aperture.

Diffraction at a circular aperture

The field strength behind an illuminated with monochromatic light pinhole follows function

The distance from the point of maximum intensity and the Bessel function of the first kind.

The light intensity of ~ follows function

The intensity goes to zero at regular intervals and contains outward weakening secondary maxima. The size of the central diffraction disk results from the first zero of the function, the ... is at r = 0.6098.

The angular diameter of the rim of the central diffraction disk is found to be out of the double angle radius:

With

• = Wavelength of light and
• = Diameter of the aperture.

• For r = 0 has the function value of 1,
• The standard deviation of the first diffraction disk is 0.1975,
• R = 0.2572 has the function value √ ½ (50% intensity )
• R = 0.3526 has the function value ½ (25% intensity )
• Has at r = 0.6098 the first zero
• Has at r = 0.8173 the first secondary maximum with the function value -0.1322 (1.75 % of the intensity )
• Has more zeros at 1.1166, 1.6192, ...
• Has more secondary maxima at 1.3396, 1.8493, ...
• Obtained by Excel ( BESSELJ (2 * PI () * x 1) / ( PI () * x) ) ^ 2 with the argument x

The size of the diffraction disk, resulting from the effective aperture diameter of an optical system, determines the resolution. Two points can then safely disconnect ( according to the Rayleigh criterion ) when the maxima of their images are at least the radius of the diffraction disk apart.

Forms a lens from infinity to the focal length from, the central diffraction disk has diameter

With

• = Wavelength of light
• = Focal length of the imaging lens,
• = Diameter of the lens, and
• = F-number

The larger the diameter, or the smaller the F number, the smaller the angle or the diameter of the diffraction disk. Therefore, high-resolution telescopes need large mirror.

Approximate formula for estimating

In practice, one often expects the following approximation formulas:

Other panel models

Deviates from the aperture of the circular shape, the shape of the central maximum and the higher diffraction orders changed. The picture on the left shows an example of a rectangular aperture. Your orientation is indicated in the upper left corner of the image. The ratio of height and width is also reflected in the central spot resist, but with reciprocal relationships, as aperture and diffraction pattern are linked via the Fourier transform. The secondary maxima are most pronounced in the principal directions.

The picture on the right shows the Airy disk (right ) of different aperture (left). The annular intensity modulation, one would expect at a circular aperture is superimposed by radial stars, the so-called spikes. Particularly clear they emerge in the triangle aperture.

If a dark visor used, resulting in the shade of the corresponding circular disc also a typical diffraction image with a Poisson's spot in the middle.

Examples of diffraction-limited resolution

All considerations occur unless otherwise specified, at a wavelength of 555 nm (green).

• A diffraction-limited lens with an f-number of 2.95 produces an Airy disk diameter of 4 microns. This can be seen on extremely high-resolution black -and-white negative films. In an f-number of 11, the Airy disk already has a diameter of 15 microns and can be seen on many medium -resolution film emulsions. The same applies for a digital sensor with a pixel pitch of 6 microns.

• When the International Space Station is equipped with a lens with 14 cm diameter, details of the size of 1 can " dissolve. At an altitude of 350 km which corresponds to a resolution of 1.7 m. To photograph these details, they must be larger than the resolution of the sensor. If this is 4.8 microns, a focal length of at least 1 m is required.

• The Hubble Space Telescope orbits the Earth at an altitude of 590 km. His mirror has a diameter of 240 cm. Directed to the earth it would have under optimal conditions a resolution of 0.14 m.

• Large mirrors are expensive. Spy satellites compensate the disadvantage of smaller mirror with a low altitude. With a mirror diameter of 100 cm and an altitude of 150 km, a resolution of 0.10 m is theoretically possible. With drones, leading to lower altitudes despite even smaller lenses, the achievable resolution is even higher.

• A commonplace example of observable diffraction disk is the intrinsic perception, ie the perception of a stimulus, which has its origins in the involved in the perception of sensory organ itself, which is caused by dead cells in the aqueous humor of the eye. If you look for a bright, solid color area as possible (for example, against the sky ) so you can see faint, transparent Squiggle, which slowly sink to the bottom and often join together to form chains: just the Airy disk, which by mentioned are caused in the aqueous humor floating cells.