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Encyclopedia > Diffraction
The intensity pattern formed on a screen by diffraction from a square aperture

Diffraction refers to various phenomena associated with wave propagation, such as the bending, spreading and interference of waves passing by an object or aperture that disrupts the wave. It occurs with any type of wave, including sound waves, water waves, electromagnetic waves such as visible light, x-rays and radio waves. Diffraction also occurs with matter – according to the principles of quantum mechanics, any physical object has wave-like properties. While diffraction always occurs, its effects are generally most noticeable for waves where the wavelength is on the order of the feature size of the diffracting objects or apertures. The complex patterns in the intensity of a diffracted wave are a result of interference between different parts of a wave that traveled to the observer by different paths. Image File history File links Square_diffraction. ... Image File history File links Square_diffraction. ... Interference of two circular waves - Wavelength (decreasing bottom to top) and Wave centers distance (increasing to the right). ... A wave is a disturbance that propagates through space or spacetime, transferring energy and momentum and sometimes angular momentum. ... Sound is a disturbance of mechanical energy that propagates through matter as a wave. ... Impact from a water drop causes an upward rebound jet surrounded by circular capillary waves. ... Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. ... The optical spectrum (light or visible spectrum) is the portion of the electromagnetic spectrum that is visible to the human eye. ... In the NATO phonetic alphabet, X-ray represents the letter X. An X-ray picture (radiograph) taken by Röntgen An X-ray is a form of electromagnetic radiation with a wavelength approximately in the range of 5 pm to 10 nanometers (corresponding to frequencies in the range 30 PHz... Radio waves are electromagnetic waves occurring on the radio frequency portion of the electromagnetic spectrum. ... Fig. ... A wave is a disturbance that propagates through space or spacetime, transferring energy and momentum and sometimes angular momentum. ...

## Examples of diffraction in everyday life GA_googleFillSlot("encyclopedia_square");

Diffraction can occur with any kind of wave. Ocean waves diffract around jetties and other obstacles. Sound waves can diffract around objects, this is the reason we can still hear someone calling us even if we are hiding behind a tree. Diffraction can also be a concern in some technical applications; it sets a fundamental limit to the resolution of a camera, telescope, or microscope. The ability to produce optical images with angular separations as small as the instruments theoretical limit. ...

## History

Thomas Young's sketch of two-slit diffraction, which he presented to the Royal Society in 1803

The effects of diffraction of light were first carefully observed and characterized by Francesco Maria Grimaldi, who also coined the term diffraction, from the Latin diffringere, 'to break into pieces', referring to light breaking up into different directions. The results of Grimaldi's observations were published posthumously in 1665.[1][2] Isaac Newton studied these effects and attributed them to inflexion of light rays. James Gregory (1638–1675) observed the diffraction patterns caused by a bird feather, which was effectively the first diffraction grating. In 1803 Thomas Young did his famous experiment observing diffraction from two closely spaced slits. Explaining his results by interference of the waves emanating from the two different slits, he deduced that light must propagate as waves. Augustin-Jean Fresnel did more definitive studies and calculations of diffraction, published in 1815 and 1818, and thereby gave great support to the wave theory of light that had been advanced by Christian Huygens and reinvigorated by Young, against Newton's particle theory. Image File history File links Young_Diffraction. ... Image File history File links Young_Diffraction. ... The premises of The Royal Society in London (first four properties only). ... Francesco Maria Grimaldi (April 2, 1618 - December 28, 1663) was an Italian mathematician and physicist who taught at the Jesuit college in Bologna. ... Sir Isaac Newton FRS (4 January 1643 â€“ 31 March 1727) [ OS: 25 December 1642 â€“ 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... James Gregory For other people with the same name, see James Gregory. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Thomas Young, English scientist Thomas Young (June 13, 1773-May 10, 1829) was an English polymath, contributing to the scientific understanding of vision, light, solid mechanics, energy, physiology, and Egyptology. ... Augustin Fresnel Augustin-Jean Fresnel (pronounced [] in AmE (or fray-NELL), [] in French) (May 10, 1788 â€“ July 14, 1827), was a French physicist who contributed significantly to the establishment of the theory of wave optics. ... Christiaan Huygens Christiaan Huygens (approximate pronunciation: HOW-khens; SAMPA /h9yGEns/ or /[email protected]@ns/) (April 14, 1629&#8211;July 8, 1695), was a Dutch mathematician and physicist; born in The Hague as the son of Constantijn Huygens. ...

## The machanism of diffraction

Photograph of single-slit diffraction in a circular ripple tank

The most conceptually simple example of diffraction is single-slit diffraction in which the slit is narrow, that is, significantly smaller than a wavelength of the wave. After the wave passes through the slit a pattern of semicircular ripples is formed, as if there were a simple wave source at the position of the slit. This semicircular wave is a diffraction pattern. The wavelength is the distance between repeating units of a wave pattern. ...

If we now consider two such narrow apertures, the two radial waves emanating from these apertures can interfere with each other. Consider for example, a water wave incident on a screen with two small openings. The total displacement of the water on the far side of the screen at any point is the sum of the displacements of the individual radial waves at that point. Now there are points in space where the wave emanating from one aperture is always in phase with the other, i.e. they both go up at that point, this is called constructive interference and results in a greater total amplitude. There are also points where one radial wave is out of phase with the other by one half of a wavelength, this would mean that when one is going up, the other is going down, the resulting total amplitude is decreased, this is called destructive interference. The result is that there are regions where there is no wave and other regions where the wave is amplified.

Another conceptually simple example is diffraction of a plane wave on a large (compared to the wavelength) plane mirror. The only direction at which all electrons oscillating in the mirror are seen oscillating in phase with each other is the specular (mirror) direction – thus a typical mirror reflects at the angle which is equal to the angle of incidence of the wave. This result is called the law of reflection. Smaller and smaller mirrors diffract light over a progressively larger and larger range of angles. Diagram of specular reflection Specular reflection is the perfect, mirror-like reflection of light from a surface, in which light from a single incoming direction is reflected onto a single outgoing direction. ... The reflection of sunlight on water Reflection is the abrupt change in direction of a wave front at an interface between two dissimilar media so that the wave front returns into the medium from which it originated. ...

Slits significantly wider than a wavelength will also show diffraction which is most noticeable near their edges. The center part of the wave shows limited effects at short distances, but exhibits a stable diffraction pattern at longer distances. This pattern is most easily understood and calculated as the interference pattern of a large number of simple sources spaced closely and evenly across the width of the slit.

This concept is known as the Huygens–Fresnel principle: The propagation of a wave can be visualized by considering every point on a wavefront as a point source for a secondary radial wave. The subsequent propagation and interference of all these radial waves form the new wavefront. This principle mathematically results from interference of waves along all allowed paths between the source and the detection point (that is, all paths except those that are blocked by the diffracting objects). Wave refraction in the manner of Huygens. ...

## Qualitative observations of diffraction

Several qualitative observations can be made of diffraction in general:

• The angular spacing of the features in the diffraction pattern is inversely proportional to the dimensions of the object causing the diffraction, in other words: the smaller the diffracting object the 'wider' the resulting diffraction pattern and vice versa. (More precisely, this is true of the sines of the angles.)
• The diffraction angles are invariant under scaling; that is, they depend only on the ratio of the wavelength to the size of the diffracting object.
• When the diffracting object has a periodic structure, for example in a diffraction grating, the features generally become sharper. The third figure, for example, shows a comparison of a double-slit pattern with a pattern formed by five slits, both sets of slits having the same spacing, between the center of one slit and the next.

In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... Double-slit diffraction and interference pattern The double-slit experiment consists of letting light diffract through two slits, which produces fringes or wave-like interference patterns on a screen. ...

## Quantitative description of diffraction

For more details on this topic, see Diffraction formalism.

To determine the pattern produced by diffraction we must determine the phase and amplitude of each of the Huygens wavelets at each point in space. That is, at each point in space, we must determine the distance to each of the simple sources on the incoming wavefront. If the distance to each of the simple sources differs by an integer number of wavelengths, all the wavelets will be in phase, resulting in constructive interference. If the distance to each source is an integer plus one half of a wavelength, there will be complete destructive interference. Usually it is sufficient to determine these minimums and maximums to explain the effects we see in nature. The simplest descriptions of diffraction are those in which the situation can be reduced to a 2 dimensional problem. For water waves, this is already the case, water waves propagate only on the surface of the water. For light, we can often neglect one direction if the diffracting object extends in that direction over a distance far greater than the wavelength. In the case of light shining through small circular holes we will have to take into account the full three dimensional nature of the problem. // To determine the pattern produced by diffraction we must determine the phase and amplitude of each of the Huygens wavelets at each point in space. ...

### Diffraction from an array of narrow slits or a grating

Diagram of two slit diffraction problem, showing the angle to the first minimum, where a path length difference of a half wavelength causes destructive interference.

Multiple-slit arrangements can be described as multiple simple wave sources, if the slits are narrow enough. For light, a slit is an opening that is infinitely extended in one dimension, which has the effect of reducing a wave problem in 3-space to a simpler problem in 2-space. The simplest case is that of two narrow slits, spaced a distance d apart. To determine the maxima and minima in the amplitude we must determine the difference in path length to the first slit and to the second one. In the Fraunhofer approximation, with the observer far away from the slits, the difference in path length to the two slits can be seen from the image to be To meet Wikipedias quality standards, this article or section may require cleanup. ... Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ...

ΔS = asinθ

Maxima in the intensity occur if this path length difference is an integer number of wavelengths.

 asinθ = nλ where n is an integer that labels the order of each maximum, λ is the wavelength, a is the distance between the slits and θ is the angle at which constructive interference occurs

And the corresponding minima are at path differences of an integer number plus one half of the wavelength: The integers are commonly denoted by the above symbol. ...

${a} sin theta = lambda (n+1/2) ,$

For an array of slits, positions of the minima and maxima are not changed, the fringes visible on a screen however do become sharper as can be seen in the image. The same is true for a surface that is only reflective along a series of parallel lines; such a surface is called a reflection grating.

2-slit and 5-slit diffraction of red laser light

We see from the formula that the diffraction angle is wavelength dependent. This means that different colors of light will diffract in different directions, which allows us to separate light into its different color components. Gratings are used in spectroscopy to determine the properties of atoms and molecules, as well as stars and interstellar dust clouds by studying the spectrum of the light they emit or absorb. Another application of diffraction gratings is to produce a monochromatic light source. This can be done by placing a slit at the angle corresponding to the constructive interference condition for the desired wavelength. (c) 2002 B. Crowell, GFDL licensed File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Extremely high resolution spectrogram of the Sun showing thousands of elemental absorption lines (fraunhofer lines) Spectroscopy is the study of the interaction between radiation (electromagnetic radiation, or light, as well as particle radiation) and matter. ...

### Single-slit diffraction

Numerical approximation of diffraction pattern from a slit of width four wavelengths with an incident plane wave. The main central beam, nulls, and phase reversals are apparent.
Graph and image of single-slit diffraction

Slits wider than a wavelength will show diffraction at their edges. The pattern is most easily understood and calculated as the interference pattern of a large number of simple sources spaced closely and evenly across the width of the slit. We can determine the minima of the resulting intensity pattern by using the following reasoning. If for a given angle a simple source located at the left edge of the slit interferes destructively with a source located at the middle of the slit, then a simple source just to the right of the left edge will interfere destructively with a simple source located just to the right of the middle. We can continue this reasoning along the entire width of the slit to conclude that the condition for destructive interference for the entire slit is the same as the condition for destructive interference between two narrow slits a distance apart that is half the width of the slit. The result is a formula that looks very similar to the one for diffraction from a grating with the important difference that it now predicts the minima of the intensity pattern. Image File history File links Wave_Diffraction_4Lambda_Slit. ... Image File history File links Wave_Diffraction_4Lambda_Slit. ... from meta File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... from meta File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...

dsin(θmin) = nλ n is now an integer greater than 0.

The same argument does not hold for the maxima. To determine the location of the maxima and the exact intensity profile, a more rigorous treatment is required; a diffraction formalism in terms of integration over all unobstructed paths is required. The intensity profile is then given by // To determine the pattern produced by diffraction we must determine the phase and amplitude of each of the Huygens wavelets at each point in space. ...

 $I(theta),$ $= I_0 {left[ operatorname{sinc} left( frac{pi a}{lambda} sin theta right) right] }^2$

### Multiple extended slits

For an array of slits that are wider than the wavelength of the incident wave, we must take into account interference of wave from different slits as well as interference between waves from different locations in the same slit. Minima in the intensity occur if either the single slit condition or the grating condition for complete destructive interference is met. A rigorous mathematical treatment shows that the resulting intensity pattern is the product of the grating intensity function with the single slit intensity pattern.

$Ileft(thetaright) = I_0 left[ operatorname{sinc} left( frac{pi a}{lambda} sin theta right) right]^2 cdot left[frac{sinleft(frac{Npi d}{lambda}sinthetaright)}{sinleft(frac{pi d}{lambda}sinthetaright)}right]^2$

When doing experiments with gratings that have a slit width being an integer fraction of the grating spacing, this can lead to 'missing' orders. If for example the width of a single slit is half the separation between slits, the first minimum of the single slit diffraction pattern will line up with the first maximum of the grating diffraction pattern. This expected diffraction peak will then not be visible. The same is true in this case for any odd numbered grating-diffraction peak.

## Particle diffraction

Quantum theory tells us that every particle exhibits wave properties. In particular, massive particles can interfere and therefore diffract. Diffraction of electrons and neutrons stood as one of the powerful arguments in favor of quantum mechanics. The wavelength associated with a particle is the de Broglie wavelength Neutron diffraction is a crystallography technique that uses neutrons to determine the atomic structure of a material. ... This article is in need of attention from an expert on the subject. ... Fig. ... In physics, the de Broglie hypothesis is the statement that all matter (any object) has a wave-like nature (wave-particle duality). ...

$lambda=frac{h}{p}$

where h is Planck's constant and p is the momentum of the particle (mass × velocity for slow-moving particles) . For most macroscopic objects, this wavelength is so short that it is not meaningful to assign a wavelength to them. A Sodium atom traveling at about 3000 m/s would have a De Broglie wavelength of about 5 pico meters. A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... In classical mechanics, momentum (pl. ...

Because the wavelength for even the smallest of macroscopic objects is extremely small, diffraction of matter waves is only visible for small particles, like electrons, neutrons, atoms and small molecules. The short wavelength of these matter waves makes them ideally suited to study the atomic crystal structure of solids and large molecules like proteins.

Relatively recently, larger molecules like buckyballs,[3] have been shown to diffract. Currently research is underway into the diffraction of viruses, which, being huge relative to electrons and other more commonly diffracted particles, have tiny wavelengths so must be made to travel very slowly through an extremely narrow slit in order to diffract. It is believed that there is a limit to the largest object that can be diffracted. Fullerene C540 Fullerenes are one of only 3 types of naturally occurring forms of carbon (the other two being diamond and graphite). ... Stop editing pages god ... Properties The electron (also called negatron, commonly represented as e&#8722;) is a subatomic particle. ...

## Bragg diffraction

For more details on this topic, see Bragg diffraction.

Diffraction from a three dimensional periodic structure such as atoms in a crystal is called Bragg diffraction. It is similar to what occurs when waves are scattered from a diffraction grating. Bragg diffraction is a consequence of interference between waves reflecting from different crystal planes. The condition of constructive interference is given by Bragg's law: The Bragg formulation of X-ray diffraction (also referred to as Bragg diffraction) was first proposed by William Lawrence Bragg and William Henry Bragg in 1913 in response to their discovery that crystalline solids produced surprising patterns of reflected X-rays (in contrast to that of, say, a liquid). ... The Bragg formulation of X-ray diffraction (also referred to as Bragg diffraction) was first proposed by William Lawrence Bragg and William Henry Bragg in 1913 in response to their discovery that crystalline solids produced surprising patterns of reflected X-rays (in contrast to that of, say, a liquid). ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

mλ = 2dsinθ

where

λ is the wavelength,
d is the distance between crystal planes,
θ is the angle of the diffracted wave.
and m is an integer known as the order of the diffracted beam.

Bragg diffraction may be carried out using either light of very short wavelength like x-rays or matter waves like neutrons whose wavelength is on the order of the atomic spacing. The pattern produced gives information of the separations of crystallographic planes d, allowing one to deduce the crystal structure. The wavelength is the distance between repeating units of a wave pattern. ... X-ray crystallography is a technique in crystallography in which the pattern produced by the diffraction of x-rays through the closely spaced lattice of atoms in a crystal is recorded and then analyzed to reveal the nature of that lattice. ... Neutron diffraction is a crystallography technique that uses neutrons to determine the atomic structure of a material. ...

## Coherence

Main article: Coherence (physics)

The description of diffraction relies on the interference of waves emanating from the same source taking different paths to the same point on a screen. In this description, the difference in phase between waves that took different paths is only dependent on the effective path length. This does not take into account the fact that waves that arrive at the screen at the same time were emitted by the source at different times. The initial phase with which the source emits waves can change over time in an unpredictable way. This means that waves emitted by the source at times that are too far apart can no longer form a constant interference pattern since the relation between their phases is no longer time independent. Coherence is the property of wave-like states that enables them to exhibit interference. ...

The length over which the phase in a beam of light is correlated, is called the coherence length. In order for interference to occur, the path length difference must be smaller than the coherence length. This is sometimes referred to as spectral coherence as it is related to the presence of different frequency components in the wave. In the case light emitted by an atomic transition, the coherence length is related to the lifetime of the excited state from which the atom made its transition. Coherence is the property of wave-like states that enables them to exhibit interference. ... A quantum mechanical system can only be in certain states, so that only certain energy levels are possible. ...

If waves are emitted from an extended source this can lead to incoherence in the transversal direction. When looking at a cross section of a beam of light, the length over which the phase is correlated is called the transverse coherence length. In the case of Young's double slit experiment this would mean that if the transverse coherence length is smaller than the spacing between the two slits the resulting pattern on a screen would look like two single slit diffraction patterns.

In the case of particles like electrons, neutrons and atoms, the coherence length is related to the spacial extent of the wave function that describes the particle.

## Diffraction limit of telescopes

The Airy disc around each of the stars from the 2.56m telescope aperture can be seen in this lucky image of the binary star zeta Boötis.

For diffraction through a circular aperture, there is a series of concentric rings surrounding a central Airy disc. The mathematical result is similar to a radially symmetric version of the equation given above in the case of single-slit diffraction. Wikipedia does not have an article with this exact name. ... A computer generated image of an Airy Disc. ... Lucky image of M15 core Lucky imaging is an astronomical photographic technique using a high-speed camera with exposure times short enough (100ms or less) so that the changes in the atmosphere during the exposure are minimal. ... Artists impression of a binary system consisting of a black hole, with an accretion disc around it, and a main sequence star. ... Zeta Bootis imaged with the Nordic Optical Telescope on 13 May 2000 using the lucky imaging method. ... A computer generated image of an Airy Disc. ...

A wave does not have to pass through an aperture to diffract; for example, a beam of light of a finite size also undergoes diffraction and spreads in diameter. This effect limits the minimum diameter d of spot of light formed at the focus of a lens, known as the diffraction limit: DIAMETER is a computer networking protocol for AAA (Authentication, Authorization and Accounting). ... A lens. ... The ability to produce optical images with angular separations as small as the instruments theoretical limit. ...

$d = 1.22 lambda frac{f}{a},,$

where λ is the wavelength of the light, f is the focal length of the lens, and a is the diameter of the beam of light, or (if the beam is filling the lens) the diameter of the lens. The diameter given is enough to contain about 70% of the light energy; it is the radius to the first null of the Airy disk, in approximate agreement with the Rayleigh criterion. Twice that diameter, the diameter to the first null of the Airy disk, within which 83.8% of the light energy is contained, is also sometimes given as the diffraction spot diameter. Categories: Optics | Science stubs ... Resolving power is the ability of a microscope or telescope to measure the angular separation of images that are close together. ... Categories: Optics | Science stubs ...

By use of Huygens' principle, it is possible to compute the diffraction pattern of a wave from any arbitrarily shaped aperture. If the pattern is observed at a sufficient distance from the aperture, it will appear as the two-dimensional Fourier transform of the function representing the aperture. Wave Refraction in the manner of Huygens. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ...

## References

1. ^ Jean Louis Aubert (1760). Memoires pour l'histoire des sciences et des beaux arts. Paris: Impr. de S. A. S.; Chez E. Ganeau, 149.
2. ^ Sir David Brewster (1831). A Treatise on Optics. London: Longman, Rees, Orme, Brown & Green and John Taylor, 95.
3. ^ Brezger, B.; Hackermüller, L.; Uttenthaler, S.; Petschinka, J.; Arndt, M.; Zeilinger, A. (February 2002). "Matter-Wave Interferometer for Large Molecules" (reprint). Physical Review Letters 88 (10): 100404. DOI:10.1103/PhysRevLett.88.100404. Retrieved on 2007-04-30.

A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era. ... is the 120th day of the year (121st in leap years) in the Gregorian calendar. ...

If a shaft of light entering a prism is sufficiently narrow, a spectrum results. ... // To determine the pattern produced by diffraction we must determine the phase and amplitude of each of the Huygens wavelets at each point in space. ... External links http://www. ... The Bragg formulation of X-ray diffraction (also referred to as Bragg diffraction) was first proposed by William Lawrence Bragg and William Henry Bragg in 1913 in response to their discovery that crystalline solids produced surprising patterns of reflected X-rays (in contrast to that of, say, a liquid). ... To meet Wikipedias quality standards, this article or section may require cleanup. ... This article is in need of attention from an expert on the subject. ... Neutron diffraction is a crystallography technique that uses neutrons to determine the atomic structure of a material. ... X-ray crystallography is a technique in crystallography in which the pattern produced by the diffraction of x-rays through the closely spaced lattice of atoms in a crystal is recorded and then analyzed to reveal the nature of that lattice. ... The dynamical theory of diffraction describes the interaction of wave fields with a regular lattice. ... Fraunhofer diffraction is diffraction of light through an aperture for small values of the Fresnel number, F<<1. ... Fresnel diffraction is the diffraction pattern of an electromagnetic wave obtained a small number of wavelengths away from the diffracting object (often a source or aperture). ... The Fresnel number F, named after the physicist Augustin-Jean Fresnel, is a dimensionless number occurring in optics, in particular in diffraction theory. ... In optics and radio communications, a Fresnel zone (pronounced as FRA-nel Zone), named for physicist Augustin-Jean Fresnel, is one of a (theoretically infinite) number of a concentric ellipsoids of revolution which define volumes in the radiation pattern of a (usually) circular aperture. ... Schaefer-Bergmann diffraction [1] is the resulting diffraction pattern of light interacting with sound waves in transparent crystals or glasses. ...

Results from FactBites:

 Diffraction - definition of Diffraction in Encyclopedia (1086 words) Diffraction also occurs when any group of waves of a finite size is propagating; for example, a narrow beam of light waves from a laser must, because of diffraction of the beam, eventually diverge into a wider beam at a sufficient distance from the laser. Since the diffraction angle θ is dependent on the wavelength λ, diffaction gratings impart angular dispersion on a beam of light. The most common demonstration of Bragg diffraction is the spectrum of colors seen reflected from a compact disc: the closely-spaced tracks on the surface of the disc form a diffraction grating, and the individual wavelengths of white light are diffracted at different angles from it, in accordance with Bragg's law.
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