# Resolution limit

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## Area of ​​Expertise - optics

The resolution limit is the size of the image quality determined by the resolution of an optical device.

#### Examples:

• With the microscope, the resolution limit indicates the dimension of the smallest recognizable structure.
• In the case of a telescope, the resolution limit is the distance between two stars that can still be distinguished.
• In spectral devices, the resolution limit indicates how large the wavelength distance between two spectral lines may be so that they can still be registered as separate.

## Learning units in which the term is dealt with

### Diffraction at circular openings30 min.

#### PhysicsopticsWave optics

This learning unit is about deepening the learning units about the interference of light. Numerous phenomena and applications from nature and technology are discussed here. This section deals specifically with the limitation of resolution of optical instruments due to the occurrence of diffraction.

## Liouville equation

the Liouville equation, after Joseph Liouville, is a description of the temporal development of a physical system in statistical mechanics, in the Hamilton formalism of classical mechanics and in quantum mechanics, there also called Von Neumann equation. The Liouville equation clearly states that the volume of any subset of the phase space is preserved under a temporal development, that is, that the flow through the phase space is volume and even orientation preserving.

## Resolving power

Resolution assets, Measure for the smallest distance between two observation values ​​or observation objects, which can be safely registered separately by an observation or measuring device.

1) Optical resolution

When the light rays emanating from an object enter an optical system, diffraction occurs at the boundary of the optical device, e.g. at the edge of the entrance pupil. A distant point of light is therefore not referred to as a point (diffraction), but as a diffraction disk (Airy discs) with the radius r = 1,22 λf/D. shown, surrounded by several concentric narrow circles, which lose intensity very quickly on the outside (λ is the wavelength used, f the focal length and D. the diameter of the lens). About 84% of the total light falls on the Airy disk, 91% of the light lies within the second dark ring. If two objects are to be imaged (Fig.) Which, viewed from the lens, are an angular distance of dϕ these are considered to be resolvable if, in the case of the diffraction disks, the maximum brightness of one object comes to lie on the first minimum brightness of the other, i.e. if the two maximum brightnesses are no closer than d = 1,22 λf/D. to have (Rayleigh criterion). Because this distance is the same f dϕ is, it follows dϕ = 1,22 λ/D.. This size will be Resolution limit called it is the reciprocal of Dissolution assets U, for which therefore applies U = 1 / dϕ = 0,82 D./λ. (Both terms are often mistakenly used synonymously.) The resolution is therefore greater, the smaller the wavelength and the larger the lens diameter. These considerations apply to both the telescope and the human eye (see below). at Microscopes one differentiates that lateral resolution, i.e. the resolution for laterally extended structures, and that Deep resolution capabilitieswhich relates to the deep structure. To determine the lateral resolving power, the minimum distance between the centers of the diffraction disks is taken with d = 1,22λ/A. numbered, where A. the numerical aperture of the objective is (A. = R./f with R.: Radius of the lens, f: Focal length of the lens). Since in microscopy differences in shape and brightness significantly influence the ability of the eye to dissolve structures, a physiological factor is added k a. The practical result is that two particles can still be observed separately from one another if their distance is approx λ/ 2 is, provided that the numerical aperture has approximately the value 1. The resolution is here also equal to the reciprocal of the resolution limit d, so is U = 1/d = 0,82 · k · A./λ. With the interference microscope, the resolution is a phase resolution, since the depth structure is observed as a phase shift. The phase resolution for incident light is about 20 /λ, f & # 252r transmitted light 10 / [(n – 1)λ] (is included n the refractive index of the object). The phase resolution can only be improved at the expense of the side resolution. Since the resolution is inversely proportional to the wavelength, it can be improved by using shorter-wave light (ultraviolet microscopy, electron microscopy). In immersion microscopy, embedding the object in an optical medium with a high refractive index can increase the numerical aperture and thus the resolution.

Another important parameter is lighting. While with telescopes and with the eye one has the power of resolution for self-luminous Objects (Helmholtz's resolution assets), Abbe has the resolution assets for not self-illuminating Body examined (mainly in relation to microscopy). According to Abbe, the creation of an image is based on the diffraction of light, with the first order of diffraction being essential for the creation of an image. The image quality is diffraction-limited when the first order of diffraction no longer passes through the device without interference. & # 220 about the relationship sinϕ = λ/d for the angle of the first order of diffraction, the minimum distance between two object points to be resolved is obtained d = λ/(n sin α) = λ/A. with α as a lens aperture angle and A. as a numerical aperture. The resulting resolution U = 1/d = A./λ is thus 1.22 greater than the resolution for self-luminous objects (see above) if additional physiological correction factors are neglected. From the standpoint of modern optics, however, there is no fundamental difference between Helmholtz's and Abbe's notions of resolving power.

2) Spectral resolution

at Spectral apparatus becomes the size U = λλ as Resolution assets Are defined. It indicates the ability of the spectral apparatus to spatially separate the intensity maxima for light waves of closely spaced wavelengths after passing through the apparatus. The wavelength-selective parameters (e.g. refractive index of a prism, diffraction angle of a grating,.) Are generally not linearly dependent on the wavelength. Strictly speaking, the resolution must therefore be specified explicitly for a wavelength or a narrow wavelength interval. Two wavelengths apply to diffraction gratings λ1 and λ2 = λ1 + Δλ then as separated if the interference maximum of λ1 in a minimum of λ2 f & # 228llt (or the distance between the interference maxima is greater than their half-width). The half-width of the interference maxima of a grating decreases with higher order, so the resolution increases. It applies λλ = pk, whereby k the diffraction order and p is the number of strokes lit. While the observable lattice order is limited by the intensity of the maxima, which decreases with higher order, the resolution of a lattice can also be influenced by the number of coherently illuminated lattice gaps. A typical one Transmission grating has approx. 600 lines per millimeter. With an illuminated width of 16 cm, this results in for & # 252r p & # 8776 100,000. If one observes in the third order of diffraction, one achieves a resolution of approx. 300,000. For comparison: the wavelength spacing of the two Na-D lines is λλ & # 8776 1000. For even higher resolutions, the illuminated width can be increased even further or the distance between the lines can be reduced. The best Rowland grid have up to 1700 lines per millimeter, holographic grating sometimes even several thousand. Due to the high demands on the surface, the manufacturing effort for resolutions greater than 300,000 increases enormously.

The spectral resolution of a Prism is given by λλ = bdn/ dλ, whereby b the base length and dn/ dλ is the dispersion of the prism. According to the principles of geometric optics, one expects an infinitely high resolution regardless of the dispersion, but in fact diffraction also limits the resolution here. The consideration of prisms of the base length provides a concept of the size of the spectral resolution b = 1 cm made of flint glass and crown glass. In the optical area, flint glass has a dispersion of 1730 cm - 1, crown glass of 530 cm - 1. The two D-lines of sodium (λ1 = 589.5932 nm, λ2 = 588.9965 nm) can therefore only be separated with the flint glass prism.

3) Photographic resolution

In the photograph The resolution is a measure of the ability of a photographic layer to reproduce small details in a recognizable manner. It is usually indicated by the number of lines per millimeter of a grid that can just be recognized in the image. The resolution depends on the type and contrast of the screen, the wavelength of the radiation, the exposure, the development and, above all, the nature of the photographic layer. The resolution is limited by the reduction in contrast, the diffusion halo and the granularity. With a line screen with high contrast, one measures around 80 lines / mm for coarse-grained layers and 150-200 lines / mm for fine-grained layers. With very fine-grained and very insensitive layers, values ​​of more than 1000 lines / mm can be achieved.

4) Resolving power of the eye

The resolving power of the eye is a measure of the eye's ability to separate spatially and temporally adjacent stimuli. That spatial resolution is & # 252 via the angular distance & # 916ϕ defined, which two object points must have with respect to the center of the pupil from each other in order to produce two separate light sensations. It is limited by the diffraction of light at the edge of the pupil and the finite size of the cones and their distance in the retina (diffraction). With & # 916ϕ = 1,22 · λ/ d follows in daylight with a pupil diameter d = 3 mm and a wavelength of λ = 550 nm a resolution limit of & # 916ϕ = 48 & # 8243 (with a factor of 1.22 diffraction). Due to its structure, the human eye can only see two points separately if they are at an angle of & # 916ϕ > 1 appear (at a distance of 25 cm, the two points must be at least 0.04 mm apart). On the retina, the diffraction disk has a radius of approx. 6 ', which corresponds to the mean distance between two cones in the retina, i.e. the resolving power of the eye is fully exploited by the arrangement of the cones in the retina. The resolution is determined with the help of a Landolt ring or different sized optotypes at a constant distance.

The smallest temporal distance between two stimuli that follow one another at the same place and lead to separate sensations determines this temporal resolution limitwhich is 50 Hz for daytime vision (stroboscopic effect) and 10 Hz for twilight vision. [KB2, MG2]

Resolutions: To explain the resolution of two object points in the image of a telescope.

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Dr. Marc Hemberger, Heidelberg [MH2] (A) (19)
Dr. Sascha Hilgenfeldt, Cambridge, USA (A) (essay sonoluminescence)
Dr. Hermann Hinsch, Heidelberg [HH2] (A) (22)
Priv.-Doz. Dr. Dieter Hoffmann, Berlin [DH2] (A, B) (02)
Dr. Gert Jacobi, Hamburg [GJ] (B) (09)
Renate Jerecic, Heidelberg [RJ] (A) (28)
Prof. Dr. Josef Kallrath, Ludwigshafen [JK] (A) (04)
Priv.-Doz. Dr. Claus Kiefer, Freiburg [CK] (A) (14, 15)
Richard Kilian, Wiesbaden [RK3] (22)
Dr. Ulrich Kilian, Heidelberg [UK] (A) (19)
Thomas Kluge, Jülich [TK] (A) (20)
Dr. Achim Knoll, Karlsruhe [AK1] (A) (20)
Dr. Alexei Kojevnikov, College Park, USA [AK3] (A) (02)
Dr. Bernd Krause, Munich [BK1] (A) (19)
Dr. Gero Kube, Mainz [GK] (A) (18)
Ralph Kühnle, Heidelberg [RK1] (A) (05)
Volker Lauff, Magdeburg [VL] (A) (04)
Dr. Anton Lerf, Garching [AL1] (A) (23)
Dr. Detlef Lohse, Twente, NL (A) (essay sonoluminescence)
Priv.-Doz. Dr. Axel Lorke, Munich [AL] (A) (20)
Prof. Dr. Jan Louis, Halle (A) (essay string theory)
Dr. Andreas Markwitz, Lower Hutt, NZ [AM1] (A) (21)
Holger Mathiszik, Celle [HM3] (A) (29)
Dr. Dirk Metzger, Mannheim [DM] (A) (07)
Dr. Rudi Michalak, Dresden [RM1] (A) (23 essay low temperature physics)
Günter Milde, Dresden [GM1] (A) (12)
Helmut Milde, Dresden [HM1] (A) (09)
Marita Milde, Dresden [MM2] (A) (12)
Prof. Dr. Andreas Müller, Trier [AM2] (A) (33)
Prof. Dr. Karl Otto Münnich, Heidelberg (A) (Essay Environmental Physics)
Dr. Nikolaus Nestle, Leipzig [NN] (A, B) (05, 20)
Dr. Thomas Otto, Geneva [TO] (A) (06)
Priv.-Doz. Dr. Ulrich Parlitz, Göttingen [UP1] (A) (11)
Christof Pflumm, Karlsruhe [CP] (A) (06, 08)
Dr. Oliver Probst, Monterrey, Mexico [OP] (A) (30)
Dr. Roland Andreas Puntigam, Munich [RAP] (A) (14)
Dr. Gunnar Radons, Mannheim [GR1] (A) (01, 02, 32)
Dr. Max Rauner, Weinheim [MR3] (A) (15)
Robert Raussendorf, Munich [RR1] (A) (19)
Ingrid Reiser, Manhattan, USA [IR] (A) (16)
Dr. Uwe Renner, Leipzig [UR] (A) (10)
Dr. Ursula Resch-Esser, Berlin [URE] (A) (21)
Dr. Peter Oliver Roll, Ingelheim [OR1] (A, B) (15)
Hans-Jörg Rutsch, Walldorf [HJR] (A) (29)
Rolf Sauermost, Waldkirch [RS1] (A) (02)
Matthias Schemmel, Berlin [MS4] (A) (02)
Prof. Dr. Erhard Scholz, Wuppertal [ES] (A) (02)
Dr. Martin Schön, Konstanz [MS] (A) (14 essay special theory of relativity)
Dr. Erwin Schuberth, Garching [ES4] (A) (23)
Jörg Schuler, Taunusstein [JS1] (A) (06, 08)
Dr. Joachim Schüller, Dossenheim [JS2] (A) (10)
Richard Schwalbach, Mainz [RS2] (A) (17)
Prof. Dr. Klaus Stierstadt, Munich [KS] (B)
Dr. Siegmund Stintzing, Munich [SS1] (A) (22)
Dr. Berthold Suchan, Giessen [BS] (A) (Essay Philosophy of Science)
Cornelius Suchy, Brussels [CS2] (A) (20)
Dr. Volker Theileis, Munich [VT] (A) (20)
Prof. Dr. Stefan Theisen, Munich (A) (essay string theory)
Dr. Annette Vogt, Berlin [AV] (A) (02)
Dr. Thomas Volkmann, Cologne [TV] (A) (20)
Rolf vom Stein, Cologne [RVS] (A) (29)
Dr. Patrick Voss-de Haan, Mainz [PVDH] (A) (17)
Dr. Thomas Wagner, Heidelberg [TW2] (A) (29)
Manfred Weber, Frankfurt [MW1] (A) (28)
Dr. Martin Werner, Hamburg [MW] (A) (29)
Dr. Achim Wixforth, Munich [AW1] (A) (20)
Dr. Steffen Wolf, Berkeley, USA [SW] (A) (16)
Dr. Stefan L. Wolff, Munich [SW1] (A) (02)
Priv.-Doz. Dr. Jochen Wosnitza, Karlsruhe [JW] (A) (23)
Dr. Kai Zuber, Dortmund [KZ] (A) (19)
Dr. Werner Zwerger, Munich [WZ] (A) (20)

### Articles on the topic

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Scientists at the University of Paderborn have succeeded in developing a new method of distance measurement for systems such as GPS, the results of which are more precise than ever before. With the help of quantum physics, the team led by Leibniz Prize winner Prof. Dr. Christine Silberhorn overcome the so-called resolution limit, which z. B. ensures the familiar noise in photos.

The results have now been published in the journal "Physical Review X Quantum" (PRX Quantum). In “Physics”, the publisher's online magazine, the paper was also recognized with an expert comment - a highlight that is only given to selected publications.

Physicist Dr. Benjamin Brecht explains the problem of the resolution limit: “In laser distance measurements, a detector registers two light pulses of different brightness with a time difference. The more precise the time measurement, the more precise the length determination. As long as the time interval between the pulses is greater than the length of the pulses, everything is fine ”. According to the scientist, it becomes problematic when the pulses overlap: “Then you can no longer measure the time difference using conventional methods. This is called the “resolution limit” and the effect of photos is known. Very small structures or textures can no longer be resolved. That is the same problem - only in place instead of in time. "

According to Brecht, another challenge is to determine the different brightnesses, the time difference and the arrival time of two light pulses at the same time. But that is exactly what the scientists succeeded in doing - "with quantum-limited accuracy," as Brecht adds. Together with partners from the Czech Republic and Spain, the Paderborn physicists were able to measure these values ​​even when the pulses overlapped by 90 percent.

Brecht: “That is far beyond the resolution limit. The accuracy of the measurement is 10,000 times better. Using methods of quantum information theory, new types of measurements can be found that overcome the limitations of established methods ”.

The results could significantly improve the accuracy of applications such as LIDAR, a method for optical distance and speed measurement, and GPS in the future. But it will take some time until it is ready for the market, says Brecht.

## Video: Sword Art Online: Alicization - War of Underworld Opening FullHaruka Tomatsu - Resolution (July 2022).

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