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| [[File:Laser Interference.JPG|thumb|Diffraction pattern of red [[laser]] beam made on a plate after passing a small circular hole in another plate]]
| | It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.<br><br><br><br>Here are some common dental emergencies:<br>Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.<br><br>At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.<br><br>Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.<br><br>Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.<br><br>Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.<br><br>Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.<br><br>Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.<br><br>In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.<br><br>If you loved this post and you would like to get more data relating to [http://www.youtube.com/watch?v=90z1mmiwNS8 Washington DC Dentist] kindly visit the page. |
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| [[Image:Square diffraction.jpg|thumb|Computer generated intensity pattern formed on
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| a screen by diffraction from a square aperture.]]
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| [[File:Two-Slit Diffraction.png|thumb|Generation of an interference pattern from two-slit diffraction.]][[File:Doubleslit.gif|thumb|Computational model of an interference pattern from two-slit diffraction.]][[File:Optical diffraction pattern ( laser), (analogous to X-ray crystallography).JPG|thumb|Optical diffraction pattern ( laser), (analogous to X-ray crystallography)]]
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| [[Image:Diffraction pattern in spiderweb.JPG|thumb|Colors seen in a [[spider web]] are partially due to diffraction, according to some analyses.<ref>{{cite web|url = http://www.itp.uni-hannover.de/%7Ezawischa/ITP/spiderweb.html|title = Optical effects on spider webs|author = Dietrich Zawischa|accessdate = 2007-09-21}}</ref>]]
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| '''Diffraction''' refers to various phenomena which occur when a wave encounters an obstacle. In classical physics, the diffraction phenomenon is described as the apparent bending of waves around small obstacles and the spreading out of waves past small openings. Similar effects occur when a light wave travels through a medium with a varying [[refractive index]], or a sound wave travels through one with varying [[acoustic impedance]]. Diffraction occurs with all waves, including [[sound]] waves, [[water]] waves, and [[electromagnetic radiation|electromagnetic waves]] such as [[visible spectrum|visible light]], [[X-ray]]s and [[radio waves]]. As physical objects have wave-like properties (at the atomic level), diffraction also occurs with matter and can be studied according to the principles of [[quantum mechanics]]. Italian scientist [[Francesco Maria Grimaldi]] coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1660.<ref>Francesco Maria Grimaldi, ''Physico mathesis de lumine, coloribus, et iride, aliisque annexis libri duo'' (Bologna ("Bonomia"), Italy: Vittorio Bonati, 1665), [http://books.google.com/books?id=FzYVAAAAQAAJ&pg=PA2#v=onepage&q&f=false page 2]: <blockquote>''Original'' : Nobis alius quartus modus illuxit, quem nunc proponimus, vocamusque; diffractionem, quia advertimus lumen aliquando diffringi, hoc est partes eius multiplici dissectione separatas per idem tamen medium in diversa ulterius procedere, eo modo, quem mox declarabimus.</blockquote> <blockquote>''Translation'' : It has illuminated for us another, fourth way, which we now make known and call "diffraction" [i.e., shattering], because we sometimes observe light break up; that is, that parts of the compound [i.e., the beam of light], separated by division, advance farther through the medium but in different [directions], as we will soon show.</blockquote></ref><ref>Cajori, Florian [http://books.google.com/books?id=KZ4C-1CRtYQC&ots=c_YpkkbTpT&dq=Florian%20Cajori%20history%20of%20physics&pg=PA88 "A History of Physics in its Elementary Branches, including the evolution of physical laboratories."] MacMillan Company, New York 1899</ref>
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| [[Richard Feynman]]<ref>R. Feynman, Lectures in Physics, Vol, 1, 1963, pg. 30-1, Addison Wesley Publishing Company Reading, Mass</ref> wrote: {{quote| No-one has ever been able to define the difference between [[interference (wave propagation)|interference]] and diffraction satisfactorily. It is just a question of usage, and there is no specific, important physical difference between them.}}
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| He suggested that when there are only a few sources, say two, we call it interference, as in [[Young's slits]], but with a large number of sources, the process is labelled diffraction.
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| While diffraction occurs whenever propagating waves encounter such changes, its effects are generally most pronounced for waves whose [[wavelength]] is roughly similar to the dimensions of the diffracting objects. If the obstructing object provides multiple, closely spaced openings, a complex pattern of varying intensity can result. This is due to the superposition, or [[Interference (wave propagation)|interference]], of different parts of a wave that travels to the observer by different paths (see [[diffraction grating]]).
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| The formalism of diffraction can also describe the way in which waves of finite extent propagate in free space. For example, the expanding profile of a laser beam, the beam shape of a radar antenna and the field of view of an ultrasonic transducer can all be analyzed using diffraction equations.
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| ==Examples==
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| [[Image:Solar glory at the steam from hot spring.jpg|[[Glory (optical phenomenon)|Solar glory]] at the [[steam]] from [[hot spring]]s. A '''glory''' is an optical phenomenon produced by light backscattered (a combination of diffraction, [[reflection (physics)|reflection]] and [[refraction]]) towards its source by a cloud of uniformly sized water droplets.|thumb]]
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| The effects of diffraction are often seen in everyday life. The most striking examples of diffraction are those that involve light; for example, the closely spaced tracks on a CD or DVD act as a [[diffraction grating]] to form the familiar rainbow pattern seen when looking at a disk. This principle can be extended to engineer a grating with a structure such that it will produce any diffraction pattern desired; the [[holography|hologram]] on a credit card is an example. [[Atmospheric diffraction|Diffraction in the atmosphere]] by small particles can cause a bright ring to be visible around a bright light source like the sun or the moon. A shadow of a solid object, using light from a compact source, shows small fringes near its edges. The [[speckle pattern]] which is observed when laser light falls on an optically rough surface is also a diffraction phenomenon. When [[deli meat]] appears to be [[iridescent]], that is diffraction off the meat fibers.<ref>{{cite web|last=Arumugam|first=Nadia|title=Food Explainer: Why Is Some Deli Meat Iridescent?|url=http://www.slate.com/blogs/browbeat/2013/09/09/iridescent_deli_meat_why_some_sliced_ham_and_beef_shine_with_rainbow_colors.html|work=Slate|publisher=[[The Slate Group]]|accessdate=9 September 2013}}</ref> All these effects are a consequence of the fact that light propagates as a [[wave]].
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| Diffraction can occur with any kind of wave. Ocean waves diffract around [[jetty|jetties]] and other obstacles. Sound waves can diffract around objects, which is why one can still hear someone calling even when hiding behind a tree.<ref>
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| {{cite book
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| |title = Dynamic fields and waves of physics
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| |author = Andrew Norton
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| |publisher = CRC Press
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| |year = 2000
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| |isbn = 978-0-7503-0719-2
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| |page = 102
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| |url = http://books.google.com/?id=XRRMxjr24pwC&pg=PA102
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| }}</ref>
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| Diffraction can also be a concern in some technical applications; it sets a [[Diffraction-limited system|fundamental limit]] to the resolution of a camera, telescope, or microscope.
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| ==History==
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| [[Image:Young Diffraction.png|right|thumb|Thomas Young's sketch of two-slit diffraction, which he presented to the Royal Society in 1803.]]
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| 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.<ref>Francesco Maria Grimaldi, ''Physico-mathesis de lumine, coloribus, et iride, aliisque adnexis...'' [The physical mathematics of light, color, and the rainbow, and other things appended...] (Bologna ("Bonomia"), Italy: Vittorio Bonati, 1665), [http://books.google.com/books?id=FzYVAAAAQAAJ&pg=PA1#v=onepage&q&f=false pp. 1–11]: "Propositio I. Lumen propagatur seu diffunditur non solum directe, refracte, ac reflexe, sed etiam alio quodam quarto modo, diffracte." (Proposition 1. Light propagates or spreads not only in a straight line, by refraction, and by reflection, but also by a somewhat different fourth way: by diffraction.)</ref><ref>{{cite book|title = Memoires pour l'histoire des sciences et des beaux arts|author = Jean Louis Aubert|publisher = Impr. de S. A. S.; Chez E. Ganeau| location = Paris|year = 1760|pages = 149|url = http://books.google.com/?id=3OgDAAAAMAAJ&pg=PP151&lpg=PP151&dq=grimaldi+diffraction+date:0-1800}}</ref><ref>{{cite book|title = A Treatise on Optics|author = Sir David Brewster|year = 1831|publisher = Longman, Rees, Orme, Brown & Green and John Taylor|location = London|pages = 95|url = http://books.google.com/?id=opYAAAAAMAAJ&pg=RA1-PA95&lpg=RA1-PA95&dq=grimaldi+diffraction+date:0-1840}}</ref> [[Isaac Newton]] studied these effects and attributed them to ''inflexion'' of light rays. [[James Gregory (astronomer and mathematician)|James Gregory]] (1638–1675) observed the diffraction patterns caused by a bird feather, which was effectively the first diffraction grating to be discovered.<ref>Letter from James Gregory to John Collins, dated 13 May 1673. Reprinted in: ''Correspondence of Scientific Men of the Seventeenth Century....'', ed. Stephen Jordan Rigaud (Oxford, England: [[Oxford University Press]], 1841), vol. 2, pp. 251–255, especially [http://books.google.com/books?id=0h45L_66bcYC&pg=PA254 p. 254].</ref> [[Thomas Young (scientist)|Thomas Young]] performed a celebrated [[Young's Double Slit Interferometer|experiment]] in 1803 demonstrating interference from two closely spaced slits.<ref>{{Cite journal|first = Thomas|last = Young|date = 1804-01-01|url = http://books.google.com/?id=7AZGAAAAMAAJ&pg=PA1|title=The Bakerian Lecture: Experiments and calculations relative to physical optics|journal=Philosophical Transactions of the [[Royal Society]] of London|volume = 94|pages = 1–16|publisher = Royal Society of London.|doi = 10.1098/rstl.1804.0001|postscript = <!--None-->}}. (Note: This lecture was presented before the Royal Society on 24 November 1803.)</ref> 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, made public in 1815<ref>Augustin-Jean Fresnel (1816) "Mémoire sur la diffraction de la lumière … ," ''Annales de la Chemie et de Physique'', 2nd series, vol. 1, pages 239–281. (Presented before ''l'Académie des sciences'' on 15 October 1815.) Available on-line at: [http://www.bibnum.education.fr/physique/optique/premier-memoire-sur-la-diffraction-de-la-lumiere Bibnum.education.fr] {{fr icon}}</ref> and 1818,<ref>Augustin-Jean Fresnel (1826) "Mémoire sur la diffraction de la lumière," ''Mémoires de l'Académie des Sciences (Paris)'', vol. 5, pages 33–475. (Summitted to ''l'Académie des sciences'' of Paris on 20 April 1818.)</ref> and thereby gave great support to the wave theory of light that had been advanced by [[Christiaan Huygens]]<ref>Christiaan Huygens, [http://books.google.com/books?id=X9PKaZlChggC&pg=PP5 ''Traité de la lumiere''...] (Leiden, Netherlands: Pieter van der Aa, 1690), Chapter 1. From [http://books.google.com/books?id=X9PKaZlChggC&pg=PA15 p. 15]: "J'ay donc monstré de quelle façon l'on peut concevoir que la lumiere s'etend successivement par des ondes spheriques,..." (I have thus shown in what manner one can imagine that light propagates successively by spherical waves,...)(Note: Huygens published his ''Traité'' in 1690; however, in the preface to his book, Huygens states that in 1678 he first communicated his book to the French Royal Academy of Sciences.)</ref> and reinvigorated by Young, against Newton's particle theory.
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| ==Mechanism==
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| [[Image:Single-slit-diffraction-ripple-tank.jpg|thumb|left|Photograph of single-slit diffraction in a circular [[ripple tank]]]]
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| Diffraction arises because of the way in which waves propagate; this is described by the [[Huygens–Fresnel principle]] and the [[superposition principle|principle of superposition of waves]]. The propagation of a wave can be visualized by considering every point on a wavefront as a point source for a secondary [[wave equation#Spherical waves|spherical wave]]. The wave displacement at any subsequent point is the sum of these secondary waves. When waves are added together, their sum is determined by the relative phases as well as the amplitudes of the individual waves so that the summed amplitude of the waves can have any value between zero and the sum of the individual amplitudes. Hence, diffraction patterns usually have a series of maxima and minima.
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| There are various analytical models which allow the diffracted field to be calculated, including the [[Kirchhoff's diffraction formula|Kirchhoff-Fresnel diffraction equation]] which is derived from [[wave equation]], the [[Fraunhofer diffraction]] approximation of the Kirchhoff equation which applies to the [[near and far field#Far_field|far field]] and the [[Fresnel diffraction]] approximation which applies to the [[near and far field#Near_field|near field]]. Most configurations cannot be solved analytically, but can yield numerical solutions through [[finite element]] and [[boundary element]] methods.
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| It is possible to obtain a qualitative understanding of many diffraction phenomena by considering how the relative phases of the individual secondary wave sources vary, and in particular, the conditions in which the phase difference equals half a cycle in which case waves will cancel one another out.
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| The simplest descriptions of diffraction are those in which the situation can be reduced to a two-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.
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| ==Diffraction of light==
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| Some examples of diffraction of light are considered below.
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| ===Single-slit diffraction===
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| {{Main|Diffraction formalism}}
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| [[File:Wavelength=slitwidthspectrum.gif|thumb|Numerical approximation of diffraction pattern from a slit of width equal to wavelength of an incident plane wave in 3D spectrum visualization]][[File:5wavelength=slitwidthsprectrum.gif|thumb|Numerical approximation of diffraction pattern from a slit of width equal to five times the wavelength of an incident plane wave in 3D spectrum visualization]]
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| [[Image:Wave Diffraction 4Lambda Slit.png|right|thumb|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.]]
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| [[Image:diffraction1.png|right|thumb|Graph and image of single-slit diffraction.]]
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| A long slit of infinitesimal width which is illuminated by light diffracts the light into a series of circular waves and the wavefront which emerges from the slit is a cylindrical wave of uniform intensity.
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| A slit which is wider than a wavelength produces interference effects in the space downstream of the slit. These can be explained by assuming that the slit behaves as though it has a large number of point sources spaced evenly across the width of the slit. The analysis of this system is simplified if we consider light of a single wavelength. If the incident light is coherent, these sources all have the same phase. Light incident at a given point in the space downstream of the slit is made up of contributions from each of these point sources and if the relative phases of these contributions vary by 2π or more, we may expect to find minima and maxima in the diffracted light. Such phase differences are caused by differences in the path lengths over which contributing rays reach the point from the slit.
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| We can find the angle at which a first minimum is obtained in the diffracted light by the following reasoning. The light from a source located at the top edge of the slit interferes destructively with a source located at the middle of the slit, when the path difference between them is equal to ''λ''/2. Similarly, the source just below the top of the slit will interfere destructively with the source located just below the middle of the slit at the same angle. We can continue this reasoning along the entire height 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 path difference is given by <math>\frac{d \sin(\theta)}{2}</math> so that the minimum intensity occurs at an angle ''θ''<sub>min</sub> given by
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| :<math>d\,\sin\theta_\text{min} = \lambda</math>
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| where
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| * ''d'' is the width of the slit,
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| * <math>\theta_\text{min}</math> is the angle of incidence at which the minimum intensity occurs, and
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| * <math>\lambda</math> is the wavelength of the light
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| A similar argument can be used to show that if we imagine the slit to be divided into four, six, eight parts, etc., minima are obtained at angles ''θ''<sub>''n''</sub> given by
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| :<math>d\,\sin\theta_{n} = n\lambda</math>
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| where
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| * ''n'' is an integer other than zero.
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| There is no such simple argument to enable us to find the maxima of the diffraction pattern. The [[diffraction formalism#Quantitative analysis of single-slit diffraction|intensity profile]] can be calculated using the [[Fraunhofer diffraction]] equation as
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| :<math>I(\theta) = I_0 \,\operatorname{sinc}^2 \left( \frac{d \pi}{\lambda} \sin\theta \right)</math>
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| where
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| * <math>I(\theta)</math> is the intensity at a given angle,
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| * <math>I_0</math> is the original intensity, and
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| * the [[unnormalized sinc function]] above is given by sinc(''x'') = sin(''x'')/(''x'') if ''x'' ≠ 0, and sinc(0) = 1
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| This analysis applies only to the [[far field]], that is, at a distance much larger than the width of the slit.
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| [[File:Diffraction2vs5.jpg|right|thumb|2-slit (top) and 5-slit diffraction of red laser light]]
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| [[File:Diffraction-red laser-diffraction grating PNr°0126.jpg|thumb|left|Diffraction of a red laser using a diffraction grating.]]
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| [[File:Diffraction 150 slits.jpg|right|thumb|A diffraction pattern of a 633 nm laser through a grid of 150 slits]]
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| ===Diffraction grating===
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| {{Main|Diffraction grating}}
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| A diffraction grating is an optical component with a regular pattern. The form of the light diffracted by a grating depends on the structure of the elements and the number of elements present, but all gratings have intensity maxima at angles θ<sub>m</sub> which are given by the grating equation
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| :<math> d \left( \sin{\theta_m} + \sin{\theta_i} \right) = m \lambda.</math>
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| where
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| * θ<sub>i</sub> is the angle at which the light is incident,
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| * ''d'' is the separation of grating elements, and
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| * ''m'' is an integer which can be positive or negative.
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| The light diffracted by a grating is found by summing the light diffracted from each of the elements, and is essentially a [[convolution]] of diffraction and interference patterns.
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| The figure shows the light diffracted by 2-element and 5-element gratings where the grating spacings are the same; it can be seen that the maxima are in the same position, but the detailed structures of the intensities are different.
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| [[Image:Airy-pattern.svg|thumb|A computer-generated image of an '''Airy disk'''.]]
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| [[Image:Airy2.gif|thumb| Computer generated light diffraction pattern from a circular aperture of diameter 0.5 micrometre at a wavelength of 0.6 micrometre (red-light) at distances of 0.1 cm – 1 cm in steps of 0.1 cm. One can see the image moving from the Fresnel region into the Fraunhofer region where the
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| Airy pattern is seen.]]
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| ===Circular aperture===
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| {{Main|Airy disk}}
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| The far-field diffraction of a plane wave incident on a circular aperture is often referred to as the [[Airy Disk]]. The [[Airy disk#Mathematical details|variation]] in intensity with angle is given by
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| :<math>I(\theta) = I_0 \left ( \frac{2 J_1(ka \sin \theta)}{ka \sin \theta} \right )^2</math>,
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| where ''a'' is the radius of the circular aperture, ''k'' is equal to 2π/λ and J<sub>1</sub> is a [[Bessel function]]. The smaller the aperture, the larger the spot size at a given distance, and the greater the divergence of the diffracted beams.
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| === General aperture ===
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| The wave that emerges from a point source has amplitude <math>\psi</math> at location r that is given by the solution of the [[frequency domain]] [[wave equation]] for a point source (The [[Helmholtz Equation]]),
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| :<math>\nabla^2 \psi + k^2 \psi = \delta(\bold r)</math>
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| where <math> \delta(\bold r)</math> is the 3-dimensional delta function. The delta function has only radial dependence, so the [[Laplace operator]] (aka scalar Laplacian) in the [[spherical coordinate system]] simplifies to (see [[del in cylindrical and spherical coordinates]])
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| :<math>\nabla ^2\psi= \frac{1}{r} \frac {\partial ^2}{\partial r^2} (r \psi) </math>
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| By direct substitution, the solution to this equation can be readily shown to be the scalar [[Green's function]], which in the [[spherical coordinate system]] (and using the physics time convention <math>e^{-i \omega t}</math>) is:
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| :<math>\psi(r) = \frac{e^{ikr}}{4 \pi r}</math>
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| This solution assumes that the delta function source is located at the origin. If the source is located at an arbitrary source point, denoted by the vector <math>\bold r'</math> and the field point is located at the point <math>\bold r</math>, then we may represent the scalar [[Green's function]] (for arbitrary source location) as:
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| :<math>\psi(\bold r | \bold r') = \frac{e^{ik | \bold r - \bold r' | }}{4 \pi | \bold r - \bold r' |}</math>
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| Therefore, if an electric field, E<sub>inc</sub>(''x'',''y'') is incident on the aperture, the field produced by this aperture distribution is given by the [[surface integral]]:
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| :<math>\Psi(r)\propto \int\!\!\!\int_\mathrm{aperture} E_{inc}(x',y')~ \frac{e^{ik | \bold r - \bold r'|}}{4 \pi | \bold r - \bold r' |} \,dx'\, dy',</math>
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| [[Image:Fraunhofer.svg|On the calculation of Fraunhofer region fields|right|thumb|350px]]
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| where the source point in the aperture is given by the vector
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| :<math>\bold{r}' = x' \bold{\hat{x}} + y' \bold{\hat{y}}</math>
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| In the far field, wherein the parallel rays approximation can be employed, the Green's function,
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| :<math>\psi(\bold r | \bold r') = \frac{e^{ik | \bold r - \bold r' |} }{4 \pi | \bold r - \bold r' |}</math>
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| simplifies to
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| :<math> \psi(\bold{r} | \bold{r}') = \frac{e^{ik r}}{4 \pi r} e^{-ik ( \bold{r}' \cdot \bold{\hat{r}})}</math>
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| as can be seen in the figure to the right (click to enlarge).
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| The expression for the far-zone (Fraunhofer region) field becomes
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| :<math>\Psi(r)\propto \frac{e^{ik r}}{4 \pi r} \int\!\!\!\int_\mathrm{aperture} E_{inc}(x',y') e^{-ik ( \bold{r}' \cdot \bold{\hat{r}} ) } \, dx' \,dy',</math>
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| Now, since
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| :<math>\bold{r}' = x' \bold{\hat{x}} + y' \bold{\hat{y}}</math>
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| and
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| :<math>\bold{\hat{r}} = \sin \theta \cos \phi \bold{\hat{x}} + \sin \theta ~ \sin \phi ~ \bold{\hat{y}} + \cos \theta \bold{\hat{z}}</math>
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| the expression for the Fraunhofer region field from a planar aperture now becomes,
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| :<math>\Psi(r)\propto \frac{e^{ik r}}{4 \pi r} \int\!\!\!\int_\mathrm{aperture} E_{inc}(x',y') e^{-ik \sin \theta (\cos \phi x' + \sin \phi y')} \, dx'\, dy'</math>
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| Letting,
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| :<math>k_x = k \sin \theta \cos \phi \,\!</math>
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| and
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| :<math>k_y = k \sin \theta \sin \phi \,\!</math>
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| the Fraunhofer region field of the planar aperture assumes the form of a [[Fourier transform]]
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| :<math>\Psi(r)\propto \frac{e^{ik r}}{4 \pi r} \int\!\!\!\int_\mathrm{aperture} E_{inc}(x',y') e^{-i (k_x x' + k_y y') } \,dx'\, dy',</math>
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| In the far-field / Fraunhofer region, this becomes the spatial [[Fourier transform]] of the aperture distribution. Huygens' principle when applied to an aperture simply says that the [[far-field diffraction pattern]] is the spatial Fourier transform of the aperture shape, and this is a direct by-product of using the parallel-rays approximation, which is identical to doing a plane wave decomposition of the aperture plane fields (see [[Fourier optics]]).
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| ===Propagation of a laser beam===
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| The way in which the [[Gaussian beam|profile]] of a [[laser|laser beam]] changes as it propagates is determined by diffraction. The output mirror of the laser is an aperture, and the subsequent beam shape is determined by that aperture. Hence, the smaller the output beam, the quicker it diverges.
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| Paradoxically, it is possible to reduce the divergence of a laser beam by first expanding it with one [[convex lens]], and then collimating it with a second convex lens whose focal point is coincident with that of the first lens. The resulting beam has a larger aperture, and hence a lower divergence.
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| ===Diffraction-limited imaging===
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| {{Main|Diffraction-limited system}}
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| [[Image:zboo lucky image 1pc.png|frame|The Airy disk around each of the stars from the 2.56 m telescope aperture can be seen in this ''[[lucky imaging|lucky image]]'' of the [[binary star]] [[zeta Boötis]].]]
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| The ability of an imaging system to resolve detail is ultimately limited by [[diffraction-limited|diffraction]]. This is because a plane wave incident on a circular lens or mirror is diffracted as described above. The light is not focused to a point but forms an [[Airy disk]] having a central spot in the focal plane with radius to first null of
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| :<math> d = 1.22 \lambda N,\, </math>
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| where λ is the wavelength of the light and ''N'' is the [[f-number]] (focal length divided by diameter) of the imaging optics. In object space, the corresponding [[angular resolution]] is
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| :<math> \sin \theta = 1.22 \frac{\lambda}{D},\, </math>
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| where ''D'' is the diameter of the [[entrance pupil]] of the imaging lens (e.g., of a telescope's main mirror).
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| Two point sources will each produce an Airy pattern – see the photo of a binary star. As the point sources move closer together, the patterns will start to overlap, and ultimately they will merge to form a single pattern, in which case the two point sources cannot be resolved in the image. The [[Rayleigh criterion]] specifies that two point sources can be considered to be resolvable if the separation of the two images is at least the radius of the Airy disk, i.e. if the first minimum of one coincides with the maximum of the other.
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| Thus, the larger the aperture of the lens, and the smaller the wavelength, the finer the resolution of an imaging system. This is why telescopes have very large lenses or mirrors, and why optical microscopes are limited in the detail which they can see.
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| ===Speckle patterns===
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| {{Main|speckle pattern}}
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| The [[speckle pattern]] which is seen when using a laser pointer is another diffraction phenomenon. It is a result of the superpostion of many waves with different phases, which are produced when a laser beam illuminates a rough surface. They add together to give a resultant wave whose amplitude, and therefore intensity varies randomly.
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| ==Patterns==
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| [[File:Diffraction on elliptic aperture with fft.png|thumb|The upper half of this image shows a diffraction pattern of He-Ne laser beam on an elliptic aperture. The lower half is its 2D Fourier transform approximately reconstructing the shape of the aperture.]]
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| Several qualitative observations can be made of diffraction in general:
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| *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 [[sine]]s of the angles.)
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| *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.
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| *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 experiment|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.
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| ==Particle diffraction==
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| {{see also|neutron diffraction|electron diffraction}}
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| 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]]
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| :<math>\lambda=\frac{h}{p} \,</math>
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| where ''h'' is [[Planck's constant]] and ''p'' is the [[momentum]] of the particle (mass × velocity for slow-moving particles).
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| 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 30,000 m/s would have a De Broglie wavelength of about 50 pico meters.
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| 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.
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| Relatively larger molecules like [[buckyballs]] were also shown to diffract.<ref>{{cite journal|journal=Physical Review Letters| title=Matter–Wave Interferometer for Large Molecules|last=Brezger|first=B.|coauthors=Hackermüller, L.; Uttenthaler, S.; Petschinka, J.; Arndt, M.; Zeilinger, A.|volume=88|issue=10|pages=100404|date=February 2002| doi=10.1103/PhysRevLett.88.100404|url=http://homepage.univie.ac.at/Lucia.Hackermueller/unsereArtikel/Brezger2002a.pdf|format=reprint|accessdate=2007-04-30|pmid=11909334|bibcode=2002PhRvL..88j0404B|arxiv = quant-ph/0202158 }}</ref>
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| ==Bragg diffraction==
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| [[Image:X-ray diffraction pattern 3clpro.jpg|thumb|Following [[Bragg's law]], each dot (or ''reflection''), in this diffraction pattern forms from the constructive interference of X-rays passing through a crystal. The data can be used to determine the crystal's atomic structure.]]
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| {{details|Bragg diffraction}}
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| Diffraction from a three dimensional periodic structure such as atoms in a crystal is called [[Bragg diffraction]].
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| 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.
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| The condition of constructive interference is given by ''Bragg's law'':
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| :<math> m \lambda = 2 d \sin \theta \,</math>
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| where
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| :λ is the wavelength,
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| :''d'' is the distance between crystal planes,
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| :θ is the angle of the diffracted wave.
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| :and ''m'' is an integer known as the ''order'' of the diffracted beam.
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| Bragg diffraction may be carried out using either light of very short wavelength like [[X-ray crystallography|X-rays]] or matter waves like [[Neutron diffraction|neutrons]] (and [[electron diffraction|electrons]]) whose wavelength is on the order of (or much smaller than) the atomic spacing.<ref>John M. Cowley (1975) ''Diffraction physics'' (North-Holland, Amsterdam) ISBN 0-444-10791-6</ref> The pattern produced gives information of the separations of crystallographic planes ''d'', allowing one to deduce the crystal structure. Diffraction contrast, in [[electron microscope]]s and [[diffraction topography|x-topography devices]] in particular, is also a powerful tool for examining individual defects and local strain fields in crystals.
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| ==Coherence==
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| {{main|Coherence (physics)}}
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| 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.
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| 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 of light emitted by an [[Energy level|atomic transition]], the coherence length is related to the lifetime of the excited state from which the atom made its transition.
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| 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.
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| In the case of particles like electrons, neutrons and atoms, the coherence length is related to the spatial extent of the wave function that describes the particle.
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| ==See also==
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| {{colbegin|3}}
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| *[[Angle-sensitive pixel]]
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| *[[Atmospheric diffraction]]
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| *[[Bragg diffraction]]
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| *[[Brocken spectre]]
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| *[[Cloud iridescence]]
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| *[[Diffraction formalism]]
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| *[[Diffraction grating]]
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| *[[Diffraction limit]]
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| *[[Diffractometer]]
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| *[[Dynamical theory of diffraction]]
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| *[[Electron diffraction]]
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| *[[Fraunhofer diffraction]]
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| *[[Fresnel diffraction]]
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| *[[Fresnel imager]]
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| *[[Fresnel number]]
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| *[[Fresnel zone]]
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| *[[Neutron diffraction]]
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| *[[Prism (optics)|Prism]]
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| *[[Powder diffraction]]
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| *[[Refraction]]
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| *[[Schaefer–Bergmann diffraction]]
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| *[[Thinned array curse]]
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| *[[X-ray scattering techniques]]
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| {{colend}}
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| ==References==
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| {{reflist|2}}
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| ==External links==
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| {{Commons|Diffraction}}
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| {{Wikibooks|Nanotechnology|Nano-optics}}
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| *[http://richannel.org/tales-from-the-prep-room-diffraction, Diffraction], Ri Channel Video, December 2011
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| *[http://www.xtal.iqfr.csic.es/Cristalografia/index-en.html Diffraction and Crystallography for beginners]
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| *[http://luminous-landscape.com/tutorials/resolution.shtml Do Sensors “Outresolve” Lenses?]; on lens and sensor resolution interaction.
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| *[http://www.acoustics.salford.ac.uk/feschools/waves/diffract.htm Diffraction and acoustics.]
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| *[http://www.johnsankey.ca/diffraction.html Diffraction in photography.]
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| *[http://www.mathpages.com/home/kmath636/kmath636.htm On Diffraction] at MathPages.
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| *[http://demonstrations.wolfram.com/search.html?query=diffraction Diffraction pattern calculators] at The Wolfram Demonstrations Project
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| *[http://www.lightandmatter.com/html_books/5op/ch05/ch05.html Wave Optics] – A chapter of an online textbook.
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| *[http://www.falstad.com/wave2d/ 2-D wave Java applet] – Displays diffraction patterns of various slit configurations.
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| *[http://www.falstad.com/diffraction/ Diffraction Java applet] – Displays diffraction patterns of various 2-D apertures.
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| *[http://www.mit.edu/~birge/diffraction/ Diffraction approximations illustrated] – MIT site that illustrates the various approximations in diffraction and intuitively explains the Fraunhofer regime from the perspective of linear system theory.
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| *[http://www.phy.hk/wiki/englishhtm/Diffraction.htm Gap] [http://www.phy.hk/wiki/englishhtm/Diffraction2.htm Obstacle] [http://www.phy.hk/wiki/englishhtm/Diffraction3.htm Corner] – Java simulation of diffraction of water wave.
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| *[http://maps.google.com/maps?q=Panama+canal&hl=en&ie=UTF8&om=1&z=16&ll=9.385048,-79.918799&spn=0.015539,0.027122&t=k&iwloc=addr Google Maps] – Satellite image of Panama Canal entry ocean wave diffraction.
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| *[http://maps.google.com/maps?f=q&source=s_q&hl=en&geocode=&q=&sll=52.788632,1.609969&sspn=0.010472,0.016093&ie=UTF8&t=h&ll=52.788217,1.606772&spn=0.010472,0.016093&z=16 Google Maps] and [http://www.bing.com/maps/?v=2&cp=52.788763840321245~1.6073888540267944&lvl=16&sty=h&eo=0 Bing Maps] – Aerial photo of waves diffracting through sea barriers at [[Sea Palling]] in [[Norfolk]], [[UK]].
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| *[http://www.cvimellesgriot.com/products/Documents/TechnicalGuide/Diffraction_Effects.pdf Diffraction Effects]
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| *[http://scripts.mit.edu/~raskar/lightfields/index.php?title=An_Introduction_to_The_Wigner_Distribution_in_Geometric_Optics An Introduction to The Wigner Distribution in Geometric Optics]
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| *[http://www.doitpoms.ac.uk/tlplib/diffraction/index.php DoITPoMS Teaching and Learning Package – Diffraction and Imaging]
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| *[http://qed.wikina.org/interference/ Animations demonstrating Diffraction] by QED
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| *[http://www.youtube.com/watch?v=uPQMI2q_vPQ FDTD Animation of single slit diffraction] on YouTube
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| [[Category:Concepts in physics]]
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| [[Category:Diffraction| ]]
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| {{Link GA|zh}}
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