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Interference (wave propagation) 1 Interference (wave propagation) Two-point interference in a ripple tank. Interference caused by the reflection on a CD In physics, interference is a phenomenon in which two waves superimpose to form a resultant wave of greater or lower amplitude. Interference usually refers to the interaction of waves that are correlated or coherent with each other, either because they come from the same source or because they have the same or nearly the same frequency. Interference effects can be observed with all types of waves, for example, light, radio, acoustic and surface water waves. Mechanism Interference of waves from two point sources. The principle of superposition of waves states that when two or more propagating waves of same type are incident on the same point, the total displacement at that point is equal to the vector sum of the displacements of the individual waves. If a crest of a wave meets a crest of another wave of the same frequency at the same point, then the magnitude of the displacement is the sum of the individual magnitudes this is constructive interference. If a crest of one wave meets a trough of another wave then the magnitude of the displacements is equal to the difference in the individual magnitudes this is known as destructive interference.

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Page 1: Interference (Wave Propagation)

Interference (wave propagation) 1

Interference (wave propagation)

Two-point interference in a rippletank.

Interference caused by the reflection on a CD

In physics, interference is a phenomenon in which two wavessuperimpose to form a resultant wave of greater or lower amplitude.Interference usually refers to the interaction of waves that arecorrelated or coherent with each other, either because they come fromthe same source or because they have the same or nearly the samefrequency. Interference effects can be observed with all types of waves,for example, light, radio, acoustic and surface water waves.

Mechanism

Interference of waves from two point sources.

The principle of superposition of waves states that when two or morepropagating waves of same type are incident on the same point, thetotal displacement at that point is equal to the vector sum of thedisplacements of the individual waves. If a crest of a wave meets acrest of another wave of the same frequency at the same point, then themagnitude of the displacement is the sum of the individual magnitudes– this is constructive interference. If a crest of one wave meets a troughof another wave then the magnitude of the displacements is equal tothe difference in the individual magnitudes – this is known asdestructive interference.

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Interference (wave propagation) 2

Magnified-image of coloured interference-patternin soap-film. The black "holes" are areas where

the film is very thin and there is near-totaldestructive-interference.

Resultant wave

Wave 1

Wave 2

Constructive interference Destructive interference

Constructive interference occurs when the phase difference between the waves is a multiple of 2π, whereasdestructive interference occurs when the difference is an odd multiple of π. If the difference between the phases isintermediate between these two extremes, then the magnitude of the displacement of the summed waves lies betweenthe minimum and maximum values.Consider, for example, what happens when two identical stones are dropped into a still pool of water at differentlocations. Each stone generates a circular wave propagating outwards from the point where the stone was dropped.When the two waves overlap, the net displacement at a particular point is the sum of the displacements of theindividual waves. At some points, these will be in phase, and will produce a maximum displacement. In other places,the waves will be in anti-phase, and there will be no net displacement at these points. Thus, parts of the surface willbe stationary—these are seen in the figure above and to the right as stationary blue-green lines radiating from thecenter.

Between two plane waves

Geometrical arrangement for two plane wave interference

A simple form of interference pattern is obtained if twoplane waves of the same frequency intersect at anangle. Interference is essentially an energyredistribution process. The energy which is lost at thedestructive interference is regained at the constructiveinterference. One wave is travelling horizontally, andthe other is travelling downwards at an angle θ to thefirst wave. Assuming that the two waves are in phase atthe point B, then the relative phase changes along thex-axis. The phase difference at the point A is given by

It can be seen that the two waves are in phase when

,

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Interference (wave propagation) 3

Interference fringes in overlapping plane waves

and are half a cycle out of phase when

Constructive interference occurs when the waves are in phase, anddestructive interference when they are half a cycle out of phase. Thus,an interference fringe pattern is produced, where the separation of themaxima is

and df is known as the fringe spacing. The fringe spacing increaseswith increase in wavelength, and with decreasing angle θ.

The fringes are observed wherever the two waves overlap and the fringe spacing is uniform throughout.

Between two spherical waves

Optical interference between two point sourcesfor different wavelengths and source separations

A point source produces a spherical wave. If the light from two pointsources overlaps, the interference pattern maps out the way in whichthe phase difference between the two waves varies in space. Thisdepends on the wavelength and on the separation of the point sources.The figure to the right shows interference between two sphericalwaves. The wavelength increases from top to bottom, and the distancebetween the sources increases from left to right.

When the plane of observation is far enough away, the fringe patternwill be a series of almost straight lines, since the waves will then bealmost planar.

Multiple beams

Interference occurs when several waves are added together providedthat the phase differences between them remain constant over the observation time.It is sometimes desirable for several waves of the same frequency and amplitude to sum to zero (that is, interferedestructively, cancel). This is the principle behind, for example, 3-phase power and the diffraction grating. In both ofthese cases, the result is achieved by uniform spacing of the phases.It is easy to see that a set of waves will cancel if they have the same amplitude and their phases are spaced equally inangle. Using phasors, each wave can be represented as for waves from to , where

.

To show that

one merely assumes the converse, then multiplies both sides by The Fabry–Pérot interferometer uses interference between multiple reflections.A diffraction grating can be considered to be a multiple-beam interferometer, since the peaks which it produces are generated by interference between the light transmitted by each of the elements in the grating. Feynman suggests that when there are only a few sources, say two, we call it "interference", as in Young's double slit experiment, but with a

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Interference (wave propagation) 4

large number of sources, the process is labelled "diffraction".[1]

Optical interference

Creation of interference fringes by an optical flat on a reflective surface.Light rays from a monochromatic source pass through the glass and reflectoff both the bottom surface of the flat and the supporting surface. The tinygap between the surfaces mean the two reflected rays have different pathlengths and interfere when they combine. At locations (b) where the pathdifference is an even multiple of λ/2, the waves reinforce. At locations (a)where the path difference is an odd multiple of λ/2 the waves cancel. Sincethe gap between the surfaces varies slightly in width at different points, a

series of alternating bright and dark bands are seen.

Because the frequency of light waves (~1014 Hz) istoo high to be detected by currently availabledetectors, it is possible to observe only theintensity of an optical interference pattern. Theintensity of the light at a given point isproportional to the square of the average amplitudeof the wave. This can be expressed mathematicallyas follows. The displacement of the two waves at apoint r is:

where A represents the magnitude of thedisplacement, φ represents the phase and ωrepresents the angular frequency.

The displacement of the summed waves is

The intensity of the light at r is given by

This can be expressed in terms of the intensities of the individual waves as

Thus, the interference pattern maps out the difference in phase between the two waves, with maxima occurring whenthe phase difference is a multiple of 2π. If the two beams are of equal intensity, the maxima are four times as brightas the individual beams, and the minima have zero intensity.The two waves must have the same polarization to give rise to interference fringes since it is not possible for wavesof different polarizations to cancel one another out or add together. Instead, when waves of different polarization areadded together, they give rise to a wave of a different polarization state.

Light source requirementsThe discussion above assumes that the waves which interfere with one another are monochromatic, i.e. have a singlefrequency—this requires that they are infinite in time. This is not, however, either practical or necessary. Twoidentical waves of finite duration whose frequency is fixed over that period will give rise to an interference patternwhile they overlap. Two identical waves which consist of a narrow spectrum of frequency waves of finite duration,will give a series of fringe patterns of slightly differing spacings, and provided the spread of spacings is significantlyless than the average fringe spacing, a fringe pattern will again be observed during the time when the two wavesoverlap.

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Interference (wave propagation) 5

Conventional light sources emit waves of differing frequencies and at different times from different points in thesource. If the light is split into two waves and then re-combined, each individual light wave may generate aninterference pattern with its other half, but the individual fringe patterns generated will have different phases andspacings, and normally no overall fringe pattern will be observable. However, single-element light sources, such assodium- or mercury-vapor lamps have emission lines with quite narrow frequency spectra. When these are spatiallyand colour filtered, and then split into two waves, they can be superimposed to generate interference fringes.[2] Allinterferometry prior to the invention of the laser was done using such sources and had a wide range of successfulapplications.A laser beam generally approximates much more closely to a monochromatic source, and it is much morestraightforward to generate interference fringes using a laser. The ease with which interference fringes can beobserved with a laser beam can sometimes cause problems in that stray reflections may give spurious interferencefringes which can result in errors.Normally, a single laser beam is used in interferometry, though interference has been observed using twoindependent lasers whose frequencies were sufficiently matched to satisfy the phase requirements.[3]

White light interference in a soap bubble

It is also possible to observe interference fringes using white light. Awhite light fringe pattern can be considered to be made up of a'spectrum' of fringe patterns each of slightly different spacing. If all thefringe patterns are in phase in the centre, then the fringes will increasein size as the wavelength decreases and the summed intensity willshow three to four fringes of varying colour. Young describes this veryelegantly in his discussion of two slit interference. Some fine examplesof white light fringes can be seen here [4]. Since white light fringes areobtained only when the two waves have travelled equal distances fromthe light source, they can be very useful in interferometry, as theyallow the zero path difference fringe to be identified.[5]

Optical arrangementsTo generate interference fringes, light from the source has to be divided into two waves which have then to bere-combined. Traditionally, interferometers have been classified as either amplitude-division or wavefront-divisionsystems.In an amplitude-division system, a beam splitter is used to divide the light into two beams travelling in differentdirections, which are then superimposed to produce the interference pattern. The Michelson interferometer and theMach-Zehnder interferometer are examples of amplitude-division systems.In wavefront-division systems, the wave is divided in space—examples are Young's double slit interferometer andLloyd's mirror.Interference can also be seen in everyday life. For example, the colours seen in a soap bubble arise from interferenceof light reflecting off the front and back surfaces of the thin soap film. Depending on the thickness of the film,different colours interfere constructively and destructively.

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Interference (wave propagation) 6

Applications of optical interferometryInterferometry has played an important role in the advancement of physics, and also has a wide range of applicationsin physical and engineering measurement.Thomas Young's double slit interferometer in 1803 demonstrated interference fringes when two small holes wereilluminated by light from another small hole which was illuminated by sunlight. Young was able to estimate thewavelength of different colours in the spectrum from the spacing of the fringes. The experiment played a major rolein the general acceptance of the wave theory of light. In quantum mechanics, this experiment is considered todemonstrate the inseparability of the wave and particle natures of light and other quantum particles (wave–particleduality). Richard Feynman was fond of saying that all of quantum mechanics can be gleaned from carefully thinkingthrough the implications of this single experiment.The results of the Michelson–Morley experiment, are generally considered to be the first strong evidence against thetheory of a luminiferous aether and in favor of special relativity.Interferometry has been used in defining and calibrating length standards. When the metre was defined as thedistance between two marks on a platinum-iridium bar, Michelson and Benoît used interferometry to measure thewavelength of the red cadmium line in the new standard, and also showed that it could be used as a length standard.Sixty years later, in 1960, the metre in the new SI system was defined to be equal to 1,650,763.73 wavelengths of theorange-red emission line in the electromagnetic spectrum of the krypton-86 atom in a vacuum. This definition wasreplaced in 1983 by defining the metre as the distance travelled by light in vacuum during a specific time interval.Interferometry is still fundamental in establishing the calibration chain in length measurement.Interferometry is used in the calibration of slip gauges (called gauge blocks in the US) and in coordinate-measuringmachines. It is also used in the testing of optical components.[6]

Radio interferometry

The Very Large Array, an interferometric array formed from manysmaller telescopes, like many larger radio telescopes.

In 1946, a technique called astronomical interferometrywas developed. Astronomical radio interferometersusually consist either of arrays of parabolic dishes ortwo-dimensional arrays of omni-directional antennas.All of the telescopes in the array are widely separatedand are usually connected together using coaxial cable,waveguide, optical fiber, or other type of transmissionline. Interferometry increases the total signal collected,but its primary purpose is to vastly increase theresolution through a process called Aperture synthesis.This technique works by superposing (interfering) thesignal waves from the different telescopes on theprinciple that waves that coincide with the same phasewill add to each other while two waves that haveopposite phases will cancel each other out. This createsa combined telescope that is equivalent in resolution (though not in sensitivity) to a single antenna whose diameter isequal to the spacing of the antennas furthest apart in the array.

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Interference (wave propagation) 7

Acoustic interferometryAn acoustic interferometer is an instrument for measuring the physical characteristics of sound waves in a gas orliquid. It may be used to measure velocity, wavelength, absorption, or impedance. A vibrating crystal creates theultrasonic waves that are radiated into the medium. The waves strike a reflector placed parallel to the crystal. Thewaves are then reflected back to the source and measured.

Quantum interference

Quantummechanics

IntroductionGlossary · History

•• v•• t• e [7]

If a system is in state , its wavefunction is described in Dirac or bra-ket notation as:

where the s specify the different quantum "alternatives" available (technically, they form an eigenvector basis)and the are the probability amplitude coefficients, which are complex numbers.The probability of observing the system making a transition or quantum leap from state to a new state is thesquare of the modulus of the scalar or inner product of the two states:

where (as defined above) and similarly are the coefficients of the final state of thesystem. * is the complex conjugate so that , etc.Now let's consider the situation classically and imagine that the system transited from to via anintermediate state . Then we would classically expect the probability of the two-step transition to be the sum ofall the possible intermediate steps. So we would have

,

The classical and quantum derivations for the transition probability differ by the presence, in the quantum case, ofthe extra terms ; these extra quantum terms represent interference between the different

intermediate "alternatives". These are consequently known as the quantum interference terms, or cross terms. This isa purely quantum effect and is a consequence of the non-additivity of the probabilities of quantum alternatives.

The interference terms vanish, via the mechanism of quantum decoherence, if the intermediate state is measuredor coupled with the environment.[8][9]

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Interference (wave propagation) 8

References[1] Richard Feynman, 1969, Lectures in Physics, Book 1, Addison Wesley, Reading, Mass.[2] WH Steel, Interferometry, 1986, Cambridge University Press, Cambridge[3] R. L. Pfleegor and L. Mandel, 1967, "Interference of independent photon beams", Phys. Rev., Volume 159, Issue 5. pp. 1084–1088.[4] http:/ / www. itp. uni-hannover. de/ ~zawischa/ ITP/ multibeam. html[5] Max Born and Emil Wolf, 1999, Principles of Optics, Cambridge University Press, Cambridge.[6] RS Longhurst, Geometrical and Physical Optics, 1968, Longmans, London.[7] http:/ / en. wikipedia. org/ w/ index. php?title=Template:Quantum_mechanics& action=edit[8] Wojciech H. Zurek, "Decoherence and the transition from quantum to classical", Physics Today, 44, pp 36–44 (1991)[9] Wojciech H. Zurek, " Decoherence, einselection, and the quantum origins of the classical (http:/ / arxiv. org/ abs/ quant-ph/ 0105127)",

Reviews of Modern Physics 2003, 75, 715.

External links• Expressions of position and fringe spacing (http:/ / www. citycollegiate. com/ interference1. htm)• Java demonstration of interference (http:/ / www. falstad. com/ ripple/ ex-2source. html)• Java simulation of interference of water waves 1 (http:/ / www. phy. hk/ wiki/ englishhtm/ Interference. htm)• Java simulation of interference of water waves 2 (http:/ / www. phy. hk/ wiki/ englishhtm/ Interference2. htm)• Flash animations demonstrating interference (http:/ / www. acoustics. salford. ac. uk/ feschools/ waves/ super2.

htm)• Lissajous Curves: Interactive simulation of graphical representations of musical intervals, beats, interference,

vibrating strings (http:/ / gerdbreitenbach. de/ lissajous/ lissajous. html)• Animations demonstrating optical interference (http:/ / qed. wikina. org/ interference/ ) by QED

Page 9: Interference (Wave Propagation)

Article Sources and Contributors 9

Article Sources and ContributorsInterference (wave propagation)  Source: http://en.wikipedia.org/w/index.php?oldid=589295035  Contributors: 4e to 4e, Ahmed1994, Aitias, Akurn, Akuyume, Aldy, Aly89, Andre Engels,Anla001, Anoko moonlight, Aranea Mortem, Army1987, Ashshydv, Atlant, Avenged Eightfold, BD2412, BigJoe Physics, Bobo192, Brews ohare, BrightStarSky, BrokenBinary, Bryan Derksen,Burgher, Cdnc, Chasingsol, Chetvorno, Chongkian, Christian75, Cleonis, Conversion script, Cookie4869, Corrigendas, CosineKitty, DJIndica, Dalcde, Danh, Deglr6328, Dekimasu, DrBob,Edcolins, Enormousdude, Epzcaw, Erguvan7, Etabackman, Ethan Mitchell, Eurion, Falcorian, Final321, Fizped, FlorianMarquardt, Furrykef, Garuh knight, Gerd Breitenbach, Giftlite, Graham87,Gurch, H Padleckas, HCPotter, HamburgerRadio, Harryboyles, Hellbus, Heron, Hyacinth, Icairns, Interiot, Iridescent, Ishanarora11 11, Ismathsadhir, Isnow, J S Lundeen, JKaver18, Jalal0,Jmcc150, JohnTechnologist, Jonathan Webley, Jordan Brown, Kaiserb, Karol Langner, KasugaHuang, KingTT, Krishnavedala, Kukini, LiborX, Lolo Sambinho, LoopyEditor, MONGO, Mac,Mandarax, Mange01, Mark Arsten, Master27, Maurice Carbonaro, Mbz1, Mejor Los Indios, Mewasul, Michael C Price, Michael Hardy, Mouse is back, Mxn, Mygerardromance, Natural Philo,Nuno Tavares, Nupsgc, Oleg Alexandrov, Omegatron, PSzalapski, Patrick0Moran, Pdelong, Pfalstad, Pflatau, Plnsaisabarish, Pol098, Possum, Prarxol, Prashanthns, Profero, Qocheedy daiin,QuantumCyclops, Quibik, Qwert, Qwertz987654321, Radiojon, Rainwarrior, Reddi, RekishiEJ, RenamedUser2, Rgoodermote, Rich Farmbrough, Rnt20, Rob-bob7-0, Royalguard11, Rp:cs,Rpetrenko, Rubikfreak, SDC, SIMMINARULA, Sandman, Saperaud, Sarsene, SchreiberBike, Sheliak, Silvarbullet1, Siw1939, Skarl the Drummer, Skizzik, Snowolf, Sodium, Solkoll, Sonett72,Srleffler, Steve Quinn, Supersam654, Sverdrup, Tangotango, The Earwig, TheAMmollusc, Theresa knott, Thingg, Tide rolls, Toytoy, Trevorcox, Tsemii, Una Smith, Unica111, Vsmith, Waldir,Waveguy, Weetoddid, William Avery, Wjh31, Wstraub, Wtshymanski, Xasthom, Zoicon5, 225 anonymous edits

Image Sources, Licenses and ContributorsImage:Two-point-interference-ripple-tank.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:Two-point-interference-ripple-tank.JPG  License: unknown  Contributors: Originaluploader was RenamedUser2 at en.wikipediaFile:Interferenz bei der Lichtreflexion an einer CD.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Interferenz_bei_der_Lichtreflexion_an_einer_CD.jpg  License: CreativeCommons Attribution-Sharealike 3.0  Contributors: User:Qwertz987654321Image:Two sources interference.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Two_sources_interference.gif  License: Public Domain  Contributors: Oleg AlexandrovFile:Interference colours in soap film 1.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Interference_colours_in_soap_film_1.jpg  License: Creative CommonsAttribution-Sharealike 3.0  Contributors: User:Natural PhiloImage:Interference of two waves.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Interference_of_two_waves.svg  License: Creative Commons Attribution-Sharealike 3.0 Contributors: original version: Haade; vectorization: Wjh31, QuibikImage:interference of plane waves 3.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Interference_of_plane_waves_3.svg  License: Creative Commons Zero  Contributors: EpzcawImage:Interferences plane waves.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Interferences_plane_waves.jpg  License: Public Domain  Contributors: Fffred, TeebeutelImage:wavepanel.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wavepanel.png  License: GNU Free Documentation License  Contributors: Bgks, Solon, TeebeutelImage:Optical flat interference.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Optical_flat_interference.svg  License: Creative Commons Zero  Contributors: User:ChetvornoImage:Soap bubble sky.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Soap_bubble_sky.jpg  License: Creative Commons Attribution-ShareAlike 3.0 Unported  Contributors:Brocken InagloryImage:USA.NM.VeryLargeArray.02.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:USA.NM.VeryLargeArray.02.jpg  License: GNU General Public License  Contributors:user:Hajor

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