Diffraction disks

Calculated diffraction image in the far field behind a circular aperture. In the representation, the intensity is implemented logarithmically on the brightness scale . This comes close to the real impression by the eye.

Diffraction disks (also: diffraction rings ) are created when a light beam is diffracted by a diaphragm . If the diaphragm is circular, a central maximum is observed, surrounded by rings of decreasing light radiation intensity . In astronomy Airy disk are also called Airy disk (Engl. Airy disc ) called, named after the English astronomer George Biddell Airy . Non-circular diaphragms also produce diffraction structures that can clearly differ from a diffraction disk ( spikes ). The diffraction of light is mathematically described by the diffraction integral .

phenomenology

Photographic diffraction image of a 90 micrometer large pinhole diaphragm with 27th diffraction orders illuminated with red laser light

Even an instrument that is perfect according to the laws of geometrical optics , without imaging errors , can not image a point of light given as an object exactly on a point; instead, the diffraction of the light at the aperture diaphragm creates a blurred spot in the image plane. The shape of the spot depends reciprocally on the shape of the aperture, in particular its size is inversely proportional to the size of the aperture. In the case of a circular aperture, given for example by the round frame of a lens, the spot is also rotationally symmetrical, with a central maximum and weak, concentric rings. Since the size of this pattern also depends on the wavelength, the diffraction rings can hardly be seen in white light. The central diffraction spot is also called the Airy disk after the English astronomer George Biddell Airy .

If one tries to keep neighboring points of an object apart and increases the magnification over the image distance, the distance between the corresponding diffraction images increases, but the diffraction images themselves also become larger in the same proportion. One speaks of diffraction limitation of the angular resolution . Diffraction images for infinite image distance are easy to calculate. This corresponds to the pinhole camera and the Fresnel approximation of the diffraction integral.

Due to the diffraction disk, which becomes smaller with the aperture on the one hand, and the aperture error , which increases with the aperture, on the other, the greatest image sharpness results in optical imaging at the critical aperture .

Diffraction at a circular aperture

Intensity behind a circular pinhole

The field strength behind a pinhole irradiated with monochromatic light follows the function ${\ displaystyle E (r)}$

{\ displaystyle E (r) = E_ {0} \ cdot {\ frac {2J_ {1} (\ pi r)} {\ pi r}}}

where is defined as follows: ${\ displaystyle r}$

{\ displaystyle r = {\ frac {kaq} {\ pi R}}}

The following applies: with the wavelength , is the radius of the aperture, represents the distance from the center of the central diffraction disk to the place where the electric field strength is to be calculated, and represents the distance between the center of the aperture and the place for which the electric field strength is to be calculated. ${\ displaystyle k = {\ tfrac {2 \ pi} {\ lambda}}}$ ${\ displaystyle \ lambda}$ ${\ displaystyle a}$ ${\ displaystyle q}$ ${\ displaystyle R}$

${\ displaystyle J_ {1} (r)}$ is the bondage function of the first kind .

The light intensity ~ follows the function ${\ displaystyle I (r)}$ ${\ displaystyle E ^ {2} (r)}$

{\ displaystyle I (r) = I_ {0} \ cdot \ left ({\ frac {2J_ {1} (\ pi r)} {\ pi r}} \ right) ^ {2}}

The intensity goes to zero at regular intervals and contains secondary maxima that become weaker towards the outside. The size of the central diffraction disk results from the first zero of the function , which is at . ${\ displaystyle 2J_ {1} (\ pi r) / (\ pi r)}$ ${\ displaystyle r = 1 {,} 2196 \ ldots}$

Diffraction pattern of a circular pinhole irradiated with white sunlight - the shorter the wavelength , the less the corresponding color components are diffracted.

{\ displaystyle \ lambda}

The angle of the edge of the central diffraction disk results from the angle radius to: ${\ displaystyle \ theta}$

{\ displaystyle \ sin \, \ theta \ approx 1 {,} 22 \ \ cdot {\ frac {\ lambda} {D}}}

and for with ${\ displaystyle \ sin \ theta = \ theta + {\ mathcal {O}} (\ theta ^ {3})}$ ${\ displaystyle \ theta \ ll 1,}$

{\ displaystyle \ theta \ approx 1 {,} 22 \ cdot {\ frac {\ lambda} {D}}}

With

${\ displaystyle \ lambda}$ = Wavelength of light and
${\ displaystyle D}$ = Diameter of the aperture.

The function

{\ displaystyle (2J_ {1} (\ pi r) / (\ pi r)) ^ {2}}

has for the function value , ${\ displaystyle r = 0}$ ${\ displaystyle 1}$
the standard deviation of the first diffraction disk is , ${\ displaystyle 0 {,} 3950}$
has for the function value (50% of the intensity), ${\ displaystyle r = 0 {,} 5145}$ ${\ displaystyle 1/2}$
has for the function value (25% of the intensity), ${\ displaystyle r = 0 {,} 7052}$ ${\ displaystyle 1/4}$
has at the first zero ${\ displaystyle r = 1 {,} 2197}$
has at the first secondary maximum with the function value (1.75% of the intensity) ${\ displaystyle r = 1 {,} 6347}$ ${\ displaystyle 0 {,} 01750}$
has further zeros at ${\ displaystyle 2 {,} 2331, \ 3 {,} 2383, \ \ ldots}$
has further secondary maxima at ${\ displaystyle 2 {,} 6793, \ 3 {,} 6987, \ \ ldots}$
can be obtained in Excel using (2*BESSELJ(PI()*x,1)/(PI()*x))^2with the argumentx

The size of the diffraction disk, which results from the effective diaphragm diameter of an optical system, determines the resolution . Two points can then be separated reliably (according to the Rayleigh criterion ) if the maxima of their images are at least apart by the radius of the diffraction disk. ${\ displaystyle r}$

If a lens images from infinity with the focal length , the central diffraction disk has the diameter ${\ displaystyle f}$ ${\ displaystyle d = 2 \ cdot \ theta \ cdot f}$

{\ displaystyle d = 2 {,} 4392 \ dots \ \ cdot {\ frac {\ lambda \ cdot f} {D}} = 2 {,} 4392 \ dots \ \ cdot \ lambda \ cdot k}

With

${\ displaystyle \ lambda}$ = Wavelength of light,
${\ displaystyle f}$ = Focal length of the imaging lens,
${\ displaystyle D}$ = Diameter of the lens and
${\ displaystyle k}$ = 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 require large mirrors. ${\ displaystyle D}$ ${\ displaystyle k = f / D}$ ${\ displaystyle \ alpha}$ ${\ displaystyle d}$

Approximation formulas for estimating

In practice, one often uses the following approximation formulas (for green light with a wavelength of 550 nm):

Diameter of the diffraction disk

{\ displaystyle d = 1 \, \ mathrm {\ mu m} \ cdot {\ tfrac {f} {D}}}

(Rayleigh criterion gives ) ${\ displaystyle d = 1 {,} 34 \, \ mathrm {\ mu m} \ cdot {\ tfrac {f} {D}}}$

Example: an aperture of results in a diffraction disk of diameter. ${\ displaystyle {\ tfrac {f} {D}} = 11}$ ${\ displaystyle 11 \, \ mathrm {\ mu m}}$

Angular resolution

{\ displaystyle \ alpha = {\ frac {100 \, \ mathrm {mm}} {D}} \ cdot {\ text {arc seconds}}}

(Rayleigh criterion gives ) ${\ displaystyle \ alpha = {\ tfrac {140 \, \ mathrm {mm}} {D}} \ cdot {\ text {arc seconds}}}$

Example: an objective aperture of diameter allows an angular resolution of . ${\ displaystyle D = 100 \, \ mathrm {mm}}$ ${\ displaystyle 1 ^ {\ prime \ prime}}$

Other aperture shapes

Left diaphragm, right calculated diffraction disks.

Detailed view: Diffraction image of a rectangular diaphragm (small picture above left), intensity of the secondary maxima exaggerated.

If the diaphragm deviates from the circular shape, the shape of the central maximum and the higher diffraction orders change. The picture on the left shows an example of a rectangular aperture. Their orientation is indicated in the upper left corner of the picture. The ratio of height and width is also reflected in the central spot, but with reciprocal ratios, since the aperture and diffraction image are linked via the Fourier transformation . The secondary maxima are most pronounced in the main directions.

The picture on the right shows diffraction disks (right) of different diaphragms (left). The ring-shaped brightness modulation that one expects from a circular aperture is superimposed by radial stars, the so-called spikes . They stand out particularly clearly with the triangular aperture.

If a dark diaphragm is used, a typical diffraction pattern with a Poisson spot in the middle is also produced in the shadow of the corresponding circular disk .

Examples of diffraction-limited resolution

Unless otherwise stated, all considerations are made at a mean wavelength in the visible range of 555 nm (green).

A diffraction-limited lens with a f-number of 3 produces a diffraction disk approximately 4 µm in diameter. This can be seen on extremely high-resolution black and white negative films in small format . At a f-number of 11, the diffraction disk already has a diameter of almost 15 µm and can be seen on many medium-resolution film emulsions. The same applies to image sensors with corresponding pixel grid spacings .
If the International Space Station ISS is equipped with a lens with a 14 cm diameter, details of the size of 1 "can be resolved. At an altitude of 350 km this corresponds to a resolution of 1.7 m. In order to be able to photograph these details, you have to greater than the resolution of the sensor, if this is 4.8 µm, a focal length of at least 1 m is required.
The Hubble Space Telescope orbits the earth at an altitude of 590 km. Its mirror has a diameter of 240 cm. Pointed to earth, it would have a resolution of 0.17 m under optimal conditions.
Large mirrors are expensive. Spy satellites compensate for the disadvantage of smaller mirrors 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 with correspondingly lower flight altitudes, the achievable resolution of objects on the earth's surface is even higher, despite even smaller lenses.
An example of diffraction disks that can be observed every day is intrinsic perception, i.e. the perception of a stimulus that originates in the sensory organ involved in the perception itself, which is caused by dead cells in the aqueous humor of the eye. If you look against a bright, preferably monochrome surface (for example the sky) you can see weak, transparent circles that slowly sink down and often join together to form chains: precisely the diffraction disks that are caused by the aforementioned cells floating in the aqueous humor.

Web links

Commons : Inflection - album containing pictures, videos and audio files

http://cnx.org/content/m13097/latest/ English (!) derivation of the intensity distribution for circular apertures (Bessel function dependency)

Individual evidence

↑ Airy discs. enzyklo.de, accessed on April 15, 2014 .
↑ The intensity of the Fraunhofer diffraction pattern of a circular aperture (the Airy pattern) is given by the squared modulus of the Fourier transform of the circular aperture: I = Io (2J1 (x) / x) ^ 2 (direct copy of the formula from the English Wikipedia did not work.)

[1] Airy discs. enzyklo.de, accessed on April 15, 2014 .

[2] The intensity of the Fraunhofer diffraction pattern of a circular aperture (the Airy pattern) is given by the squared modulus of the Fourier transform of the circular aperture: I = Io (2J1 (x) / x) ^ 2 (direct copy of the formula from the English Wikipedia did not work.)