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Wave interference

From Wikipedia, the free encyclopedia

The interference of two waves. When in phase, the two lower waves create constructive interference (left), resulting in a wave of greater amplitude. When 180° out of phase, they create destructive interference (right).
The interference of two waves. When in phase, the two lower waves create constructive interference (left), resulting in a wave of greater amplitude. When 180° out of phase, they create destructive interference (right).

In physics, interference is a phenomenon in which two waves superpose to form a resultant wave of greater, lower, or the same amplitude. Constructive and destructive interference result from 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, surface water waves, gravity waves, or matter waves. The resulting images or graphs are called interferograms.

YouTube Encyclopedic

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  • ✪ Wave interference | Mechanical waves and sound | Physics | Khan Academy
  • ✪ Interference, Reflection, and Diffraction
  • ✪ Young's double slit introduction | Light waves | Physics | Khan Academy
  • ✪ Wave Interference
  • ✪ Interference of Waves | Superposition and Interference in light and water waves | Physics


- [Voiceover] If two waves overlap in the same medium, we say that there's wave interference. So this box here could represent a speaker and this could be the sound wave it generates or it could represent a laser and this would be the light wave it generates or it could be some sort of ripple tank generator and this is the water wave it generates. Regardless, if you had a second source of a wave and these were to overlap, you'd cause wave interference and what that would look like would be something like this. So let's say these are speakers. I like thinking about it in terms of speakers, I think it's easy to think about and I put this speaker right next to the first speaker, side-by-side. So they'd be creating sound waves in this region and I wouldn't really have two sound waves necessarily. You can think of it as just having one total sound wave and how would we find the size of that total sound wave? Well if I put an axis in here. The axis will make it easier to think about this. I'm going to put an axis through here like this What I can do is I can just ask myself this, I'm going to say, what was the value of the first wave, so I'm going to take that value. What was the value of second wave and I'm just going to add them up. To get the value of the total wave, I'd take that value of the first wave plus the value of the second wave and well I'd just get double in this case. I can come over to here okay. The value here plus the value there, I get double that point. It's not going to be as high because they weren't as high.. Then over here I got zero and zero is just zero. and you start to see what's happening. I can come down here very low or very negative and I get double that down here and if I were to trace this out, what I would get is one big total sound wave that would look like this. So these have been amplified. So that's one possibility. When two waves overlap, you can get this case where the peaks match the peaks and the valleys match the valleys and you get constructive interference. So notice how each valley matches the valley, each peak matches the peak and this is called, Constructive Interference because these constructively combine to form one bigger wave. So this is Constructive Interference. So what would you hear? If your ear was over here somewhere waiting to hear this sound. What you'd actually hear is a loud note. This would be much louder than it was. It would be twice as loud in fact. Which makes sense. You've got a second speaker in here. It's twice as loud. That makes sense. What's a little bit harder to understand is you can also have something called, Destructive Interference. What would that look like? Well, imagine you had two speakers but they looked like this so that the peak of the first one lined up not with the peak of the second one but with the valley of the second one and the valley of the first one lined up with the peak of the second one. These are out of phase we say. Before, when they looked like this. These waves we say are in phase, because they look identical. The peaks match up with the peaks. The valleys with the valleys. These are out of phase. How far out of phase are they? We say that these are 180 degrees out of phase. So these are 180 degrees out of phase. The phase refers to what point on the wave cycle is the wave at and these two are starting completely separately which is 180 degrees. You might think that means 360 but think about it. If you turn around 360 degrees, you're actually back where you started. If we tried to make these 360 degrees out of phase, they'd look identical again because I've moved on so far through a cycle that's it's back to where it started in the first place. So I want to move it 180 degrees out of phase. That's exactly the opposite So that you get peak lining up with valley or if you like radions, this is called pi out of phase because pi and 180 are the same angle. Alright so what happens here if I take these two speakers? I'm going to take this second speaker and I line it up right next to the first speaker. I get something that looks more like this. Look at how weird this looks. These are completely out of phase and what's going to happen is if I add my little axis to help me think about this. I'm going to add an axis straight through here. Now I play the same game. What total wave do I end up with? Well, I take this value. I'm going to add up the values just the same. I take the value of the first wave plus the value of the second wave. I add those up, one's a positive and one's a negative I get zero and then over here zero plus zero is zero and then the valley of the first wave is lining up with the peak of the second wave and if I add these two points up, I get zero again and you probably see what's going to happen. I'm just going to get a flat line. I'm going to get a flat line and I'm going to get no wave at all. These two waves cancel and so we call this not Constructive Interference but Destructive Interference because these have destructively combined to form no wave at all and this is a little strange. How can two waves form no wave? Well, this is how you do it. And what would our ear here if we had our ear over in this area again, and we were listening. If I just had one speaker, I'd hear a noise. If I just had the second speaker, I'd hear a noise. If I have both the first and second speaker together, I don't hear anything. It's silent, which is hard to believe but this works. In fact, this is how noise canceling headphones work if you take a signal from the outside and you send in the exact same signal but flipped. Pi out of phase or 180 degrees out of phase. It cancels it and so you can fight noise with more noise but exactly out of phase and you get silence in here, or at least you can get close to it. Now you might be wondering how do we get a speaker to go 180 degrees out of phase? Well it's not too hard. If you look at the back of these speakers. Let me make a clean view. If you look at the back of these speakers, there will be a positive terminal and a negative terminal or at lease inside there will be and if you can swap the positive terminal for the negative terminal and the negative terminal for the positive terminal, then when one speaker's trying to push air forward, there's a diaphragm on this speaker moving forward and backwards. When one speaker is trying to push air forward the other speaker will be trying to pull air backwards and the net result is that the air just doesn't move because it's got equal and opposite forces on it and since the air just sits there, you've got no sound wave because air has to oscillate to create a sound wave and you get Disruptive Interference. So that's how you can create a speaker pi out of phase. You might be wondering, I don't want to mess with the wires on the back of my speaker in fact, you shouldn't so you don't get shocked but if I've got two speakers in phase like this, I'm stuck, I can't get Destructive Interference but yeah you can. Even if you don't mess with the wires, and don't, don't try this at home, you can still take this speaker, remember before when these where in phase we'd just line them up like that, Constructive Interference but I don't have to put them side-by-side. I can start one speaker a little bit forward and looks what happens. We start to get waves that are out of phase. So my question is how far forward should I move this speaker to get Destructive Interference and we can just watch. So I'm just going to try this and when we get to this point there, now we're out of phase. Now I have Destructive Interference and so how far did I move my speaker forward? If we look at it, here was the front of the speaker originally, right there. Here's the front of the speaker now. If you look at this wave, how much of a wavelength have I moved forward. The amount of wavelength that you had to move forward was 1/2 of a wavelength. So if you take two speakers that are in phase and you move one 1/2 a wavelength forward you get Destructive Interference again. Again, if my ear's over here, I'm not going to hear anything. Even though these two waves started off in phase, move one 1/2 a wavelength forward, they line up so that it's Destructive, I get no noise but if I take this away. We go back to the beginning here. Take my speaker, we start over. If you move it forward a whole wavelength, so I take this here, keep moving it, keep moving it and then Destructive Interference Whoa, here we go, Constructive Interference again. That's a whole wavelength. So if you move it forward a whole wavelength. Look, there's one whole wavelength forward. So the front of the speaker was here now the front of the speaker's here. This is an entire wavelength. I get Constructive Interference. Now I'm going to hear a loud sound again. I'm going to hear twice the noise that there would be if I just had one speaker. So the moral of this story is that even if you have speakers that are in phase, you can get Destructive Interference depending on the difference in the length that these two waves travel. In other words wave two is traveling this far to get to my ear. I'm going to call that x2 and wave one is traveling this far to get to my ear. I'm going to call that x1. If I took the difference between these two, I'd be finding the path length difference. The difference in path lengths that these waves are traveling and that would be this amount. This is the difference right here. I'm going to call it delta x because it's the magnitude of the difference between these two lengths and we saw that if this equals lambda it was constructive and if it equaled a 1/2 a lambda it was destructive but those aren't the only values. We can write down an important result here. If delta x, the path length difference was lambda or it turns out 2 lambda will work or 3 lambda, imagine moving the second speaker one more whole wavelength. Well, you'd be perfectly back in phase again because you'd align back up perfectly or 3 whole wavelengths again, perfectly in phase. Any integer wavelength including zero because zero is just the case where the speaker was right next to speaker one. Where these two speakers were lined up right next to each other, you'll get Constructive Interference. The waves line up perfectly, it's going to be constructive and we saw if, delta x equals a 1/2 wavelength it was destructive but that's not the only case. Any odd 1/2 integer here. So I can't do 2 over 2 because that would be lambda again. I could do 3 lambda over 2 or 5 lambda over 2 or 7 lambda over 2. Any of these will give me Destructive Interference because they'll cause these peaks to match up with valleys. The whole thing would flat line. I'd get no sound. This is an important result. If you've got two speakers that are starting off in phase. In other words they both start off the same way and by that I mean one speaker sends out it's wave going up, the other sends out it's wave going up. There both at the same cycle. If the only difference is the path length difference, this is an important result that let's you determine whether there's constructive or destructive interference but you might ask, hold on, what if... See this was assuming there was no phase difference to start off with. What if you did the old switcheroo on the back of one of these speakers and you swapped the positive end for the negative end so instead of coming out upward, the second one was coming out downward. Then what would happen? Well you might be able to guess. Now, this results just going to flip-flop. In other words, if I look at this case here Look at, now we start off with speakers that are out of phase to begin with. This time, if I start off with zero path length difference, I get destructive destructive instead of constructive. If I move this a whole wavelength forward, there's a whole wavelength, I get destructive again. Two wavelengths forward, destructive again. Three wavelengths forward would be destructive again and so the integer wavelength this time are going to give me destructive. What about the half integers? Let's see, I'll go forward a 1/2 a wavelength. Look at this, perfectly in phase. It's going to be constructive. How about if I go 3 1/2 of a wavelength. Again, perfectly in phase, constructive. So, in this case it turns out if you start off with speakers that were already phase shifted. If one speaker is pi shifted from the other then we got another result here. We've got that... Well actually I'll just go back to my previous result, it's easier. We can just add a little addendum here if if one speaker is pi phase shifted, from the other speaker and remember these don't have to be speakers. They could be any wave source. If one speaker's a pi phase shift from the other speaker then you just flip-flop this. Then you just take this rule and now these give you constructive right here. These would give you constructive and these up here would give you destructive and so the whole thing just gives you the opposite result. Now the whole integer wavelengths give you destructive. The 1/2 integer wavelengths give you constructive and I have to impress upon you the idea that this does not just apply for speakers. This applies for light and some sort of double slit experiment or light in a thin film experiment or sound with speakers or water waves. Any time that's the case, this rule holds in fact, this is the fundamental rule for almost all wave interference aspects. Is that the path length difference along with whether there's a pi phase shift, a relative pi phase shift between the two will determine whether you get constructive or destructive interference.



Interference of left traveling (green) and right traveling (blue) waves in one dimension, resulting in final (red) wave
Interference of left traveling (green) and right traveling (blue) waves in one dimension, resulting in final (red) wave
Interference of waves from two point sources.
Interference of waves from two point sources.
Cropped tomography scan animation of laser light interference passing through two pinholes (side edges).

The principle of superposition of waves states that when two or more propagating waves of same type are incident on the same point, the resultant amplitude at that point is equal to the vector sum of the amplitudes of the individual waves.[1] If a crest of a wave meets a crest of another wave of the same frequency at the same point, then the amplitude is the sum of the individual amplitudes—this is constructive interference. If a crest of one wave meets a trough of another wave, then the amplitude is equal to the difference in the individual amplitudes—this is known as destructive interference.

A magnified image of a coloured interference pattern in a soap film. The "black holes" are areas of almost total destructive interference (antiphase).
A magnified image of a coloured interference pattern in a soap film. The "black holes" are areas of almost total destructive interference (antiphase).

Constructive interference occurs when the phase difference between the waves is an even multiple of π (180°) , whereas destructive interference occurs when the difference is an odd multiple of π. If the difference between the phases is intermediate between these two extremes, then the magnitude of the displacement of the summed waves lies between the minimum and maximum values.

Consider, for example, what happens when two identical stones are dropped into a still pool of water at different locations. 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 the individual 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 will be stationary—these are seen in the figure above and to the right as stationary blue-green lines radiating from the centre.

Interference of light is a common phenomenon that can be explained classically by the superposition of waves, however a deeper understanding of light interference requires knowledge of wave-particle duality of light which is due to quantum mechanics. Prime examples of light interference are the famous double-slit experiment, laser speckle, anti-reflective coatings and interferometers. Traditionally the classical wave model is taught as a basis for understanding optical interference, based the Huygens–Fresnel principle.


The above can be demonstrated in one dimension by deriving the formula for the sum of two waves. The equation for the amplitude of a sinusoidal wave traveling to the right along the x-axis is

where is the peak amplitude, is the wavenumber and is the angular frequency of the wave. Suppose a second wave of the same frequency and amplitude but with a different phase is also traveling to the right

where is the phase difference between the waves in radians. The two waves will superpose and add: the sum of the two waves is

Using the trigonometric identity for the sum of two cosines: , this can be written

This represents a wave at the original frequency, traveling to the right like the components, whose amplitude is proportional to the cosine of .

  • Constructive interference: If the phase difference is an even multiple of pi: then , so the sum of the two waves is a wave with twice the amplitude
  • Destructive interference: If the phase difference is an odd multiple of π: then , so the sum of the two waves is zero

Between two plane waves

Geometrical arrangement for two plane wave interference
Geometrical arrangement for two plane wave interference
Interference fringes in overlapping plane waves
Interference fringes in overlapping plane waves

A simple form of interference pattern is obtained if two plane waves of the same frequency intersect at an angle. Interference is essentially an energy redistribution process. The energy which is lost at the destructive interference is regained at the constructive interference. One wave is travelling horizontally, and the other is travelling downwards at an angle θ to the first wave. Assuming that the two waves are in phase at the point B, then the relative phase changes along the x-axis. The phase difference at the point A is given by

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


and are half a cycle out of phase when

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

and df is known as the fringe spacing. The fringe spacing increases with 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 sources that have different wavelengths and separations of sources.
Optical interference between two point sources that have different wavelengths and separations of sources.

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

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

Multiple beams

Interference occurs when several waves are added together provided that 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, interfere destructively, cancel). This is the principle behind, for example, 3-phase power and the diffraction grating. In both of these 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 in angle. 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; see interference vs. diffraction for further discussion.

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 reflect off both the bottom surface of the flat and the supporting surface.   The tiny gap between the surfaces means the two reflected rays have different path lengths. In addition the ray reflected from the bottom plate undergoes a 180° phase reversal.  As a result, at locations (a) where the path difference is an odd multiple of λ/2, the waves reinforce.   At locations (b) where the path difference is an even multiple of λ/2 the waves cancel.  Since the gap between the surfaces varies slightly in width at different points, a series of alternating bright and dark bands, interference fringes, are seen.
Creation of interference fringes by an optical flat on a reflective surface. Light rays from a monochromatic source pass through the glass and reflect off both the bottom surface of the flat and the supporting surface. The tiny gap between the surfaces means the two reflected rays have different path lengths. In addition the ray reflected from the bottom plate undergoes a 180° phase reversal. As a result, at locations (a) where the path difference is an odd multiple of λ/2, the waves reinforce. At locations (b) where the path difference is an even multiple of λ/2 the waves cancel. Since the gap between the surfaces varies slightly in width at different points, a series of alternating bright and dark bands, interference fringes, are seen.

Because the frequency of light waves (~1014 Hz) is too high to be detected by currently available detectors, it is possible to observe only the intensity of an optical interference pattern. The intensity of the light at a given point is proportional to the square of the average amplitude of the wave. This can be expressed mathematically as follows. The displacement of the two waves at a point r is:

where A represents the magnitude of the displacement, φ represents the phase and ω represents 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 when the phase difference is a multiple of 2π. If the two beams are of equal intensity, the maxima are four times as bright as 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 waves of different polarizations to cancel one another out or add together. Instead, when waves of different polarization are added together, they give rise to a wave of a different polarization state.

Light source requirements

The discussion above assumes that the waves which interfere with one another are monochromatic, i.e. have a single frequency—this requires that they are infinite in time. This is not, however, either practical or necessary. Two identical waves of finite duration whose frequency is fixed over that period will give rise to an interference pattern while 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 significantly less than the average fringe spacing, a fringe pattern will again be observed during the time when the two waves overlap.

Conventional light sources emit waves of differing frequencies and at different times from different points in the source. If the light is split into two waves and then re-combined, each individual light wave may generate an interference pattern with its other half, but the individual fringe patterns generated will have different phases and spacings, and normally no overall fringe pattern will be observable. However, single-element light sources, such as sodium- or mercury-vapor lamps have emission lines with quite narrow frequency spectra. When these are spatially and colour filtered, and then split into two waves, they can be superimposed to generate interference fringes.[2] All interferometry prior to the invention of the laser was done using such sources and had a wide range of successful applications.

A laser beam generally approximates much more closely to a monochromatic source, and it is much more straightforward to generate interference fringes using a laser. The ease with which interference fringes can be observed with a laser beam can sometimes cause problems in that stray reflections may give spurious interference fringes which can result in errors.

Normally, a single laser beam is used in interferometry, though interference has been observed using two independent lasers whose frequencies were sufficiently matched to satisfy the phase requirements.[3] This has also been observed for widefield interference between two incoherent laser sources[4].

White light interference in a soap bubble. The iridescence is due to thin-film interference.
White light interference in a soap bubble. The iridescence is due to thin-film interference.

It is also possible to observe interference fringes using white light. A white light fringe pattern can be considered to be made up of a 'spectrum' of fringe patterns each of slightly different spacing. If all the fringe patterns are in phase in the centre, then the fringes will increase in size as the wavelength decreases and the summed intensity will show three to four fringes of varying colour. Young describes this very elegantly in his discussion of two slit interference. Since white light fringes are obtained only when the two waves have travelled equal distances from the light source, they can be very useful in interferometry, as they allow the zero path difference fringe to be identified.[5]

Optical arrangements

To generate interference fringes, light from the source has to be divided into two waves which have then to be re-combined. Traditionally, interferometers have been classified as either amplitude-division or wavefront-division systems.

In an amplitude-division system, a beam splitter is used to divide the light into two beams travelling in different directions, which are then superimposed to produce the interference pattern. The Michelson interferometer and the Mach–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 and Lloyd's mirror.

Interference can also be seen in everyday phenomena such as iridescence and structural coloration. For example, the colours seen in a soap bubble arise from interference of 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.


Optical interferometry

Interferometry has played an important role in the advancement of physics, and also has a wide range of applications in physical and engineering measurement.

Thomas Young's double slit interferometer in 1803 demonstrated interference fringes when two small holes were illuminated by light from another small hole which was illuminated by sunlight. Young was able to estimate the wavelength of different colours in the spectrum from the spacing of the fringes. The experiment played a major role in the general acceptance of the wave theory of light.[5] In quantum mechanics, this experiment is considered to demonstrate the inseparability of the wave and particle natures of light and other quantum particles (wave–particle duality). Richard Feynman was fond of saying that all of quantum mechanics can be gleaned from carefully thinking through the implications of this single experiment.[6]

The results of the Michelson–Morley experiment are generally considered to be the first strong evidence against the theory 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 the distance between two marks on a platinum-iridium bar, Michelson and Benoît used interferometry to measure the wavelength 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 the orange-red emission line in the electromagnetic spectrum of the krypton-86 atom in a vacuum. This definition was replaced 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-measuring machines. It is also used in the testing of optical components.[7]

Radio interferometry

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

In 1946, a technique called astronomical interferometry was developed. Astronomical radio interferometers usually consist either of arrays of parabolic dishes or two-dimensional arrays of omni-directional antennas. All of the telescopes in the array are widely separated and are usually connected together using coaxial cable, waveguide, optical fiber, or other type of transmission line. Interferometry increases the total signal collected, but its primary purpose is to vastly increase the resolution through a process called Aperture synthesis. This technique works by superposing (interfering) the signal waves from the different telescopes on the principle that waves that coincide with the same phase will add to each other while two waves that have opposite phases will cancel each other out. This creates a combined telescope that is equivalent in resolution (though not in sensitivity) to a single antenna whose diameter is equal to the spacing of the antennas furthest apart in the array.

Acoustic interferometry

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

Quantum interference

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 the square 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 the system. * is the complex conjugate so that , etc.

Now let's consider the situation classically and imagine that the system transited from to via an intermediate state . Then we would classically expect the probability of the two-step transition to be the sum of all 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, of the 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 is a 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 measured or coupled with its environment[clarification needed]. [8][9]

See also


  1. ^ Ockenga, Wymke. Phase contrast. Leika Science Lab, 09 June 2011. "If two waves interfere, the amplitude of the resulting light wave will be equal to the vector sum of the amplitudes of the two interfering waves."
  2. ^ WH Steel, Interferometry, 1986, Cambridge University Press, Cambridge
  3. ^ Pfleegor, R. L.; Mandel, L. (1967). "Interference of independent photon beams". Phys. Rev. 159 (5): 1084–1088. Bibcode:1967PhRv..159.1084P. doi:10.1103/physrev.159.1084.
  4. ^ Patel, R.; Achamfuo-Yeboah, S.; Light R.; Clark M. (2014). "Widefield two laser interferometry". Optics Express. 22 (22): 27094–27101.
  5. ^ a b Max Born and Emil Wolf, 1999, Principles of Optics, Cambridge University Press, Cambridge.
  6. ^ Greene, Brian (1999). The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory. New York: W.W. Norton. pp. 97–109. ISBN 0-393-04688-5.
  7. ^ RS Longhurst, Geometrical and Physical Optics, 1968, Longmans, London.
  8. ^ Wojciech H. Zurek, "Decoherence and the transition from quantum to classical", Physics Today, 44, pp 36–44 (1991)
  9. ^ Wojciech H. Zurek (2003). "Decoherence, einselection, and the quantum origins of the classical". Reviews of Modern Physics. 75: 715. arXiv:quant-ph/0105127. Bibcode:2003RvMP...75..715Z. doi:10.1103/revmodphys.75.715.

External links

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