Editing Wave interference (section)

== Mechanisms ==
[[File:Waventerference.gif|class=skin-invert-image|thumb|Interference of right traveling (green) and left traveling (blue) waves in Two-dimensional space, resulting in final (red) wave]]
[[File:Two sources interference.gif|thumb|Interference of waves from two point sources]]
[[File:3D Interference of Laser Light Through 2 Pinholes Animation.webm|thumb|thumbtime=3|Cropped tomography scan animation of laser light interference passing through two pinholes (side edges)]]

The [[superposition principle#Wave interference|principle of superposition of waves]] states that when two or more propagating waves of the same type are incident on the same point, the resultant [[amplitude]] at that point is equal to the [[Euclidean vector#Addition and subtraction|vector sum]] of the amplitudes of the individual waves.<ref>Ockenga, Wymke. [http://www.leica-microsystems.com/science-lab/phase-contrast/ 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."</ref> If a [[Crest (physics)|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'''. In ideal mediums (water, air are almost ideal) energy is always conserved, at points of destructive interference, the wave amplitudes cancel each other out, and the energy is redistributed to other areas. For example, when two pebbles are dropped in a pond, a pattern is observable; but eventually waves continue, and only when they reach the shore is the energy absorbed away from the medium.

[[File:Interference colours in soap film 1.jpg|thumb|Photograph of 1.5cm x 1cm region of soap film under white light. Varying film thickness and viewing geometry determine which colours undergo  constructive or destructive interference. Small bubbles significantly affect surrounding film thickness.]]

Constructive interference occurs when the [[Phase (waves)|phase]] difference between the waves is an '''even multiple''' of {{pi}} (180°), whereas destructive interference occurs when the difference is an '''odd multiple''' of {{pi}}. 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 unique phenomenon in that we can never observe superposition of the EM field directly as we can, for example, in water. Superposition in the EM field is an assumed phenomenon and necessary to explain how two light beams pass through each other and continue on their respective paths. Prime examples of light interference are the famous [[double-slit experiment]], [[laser speckle]], [[anti-reflective coating]]s and [[interferometer]]s. 

In addition to classical wave model for understanding optical interference, quantum [[matter waves]] also demonstrate interference. 

=== Real-valued wave functions ===
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 [[sinusoid]]al wave traveling to the right along the x-axis is
<math display="block">W_1(x,t) = A\cos(kx - \omega t)</math>
where <math>A</math> is the peak amplitude, <math>k = 2\pi/\lambda</math> is the [[wavenumber]] and <math>\omega = 2\pi f</math> 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
<math display="block">W_2(x,t) = A\cos(kx - \omega t + \varphi)</math>
where <math>\varphi</math> is the phase difference between the waves in [[radian]]s.  The two waves will [[Superposition principle|superpose]] and add: the sum of the two waves is
<math display="block">W_1 + W_2 = A[\cos(kx - \omega t) + \cos(kx - \omega t + \varphi)].</math>
Using the [[trigonometric identity]] for the sum of two cosines: <math display="inline">\cos a + \cos b = 2\cos\left({a-b \over 2}\right)\cos\left({a+b \over 2}\right),</math> this can be written
<math display="block">W_1 + W_2 = 2A\cos\left({\varphi \over 2}\right)\cos\left(kx - \omega t + {\varphi \over 2}\right).</math>
This represents a wave at the original frequency, traveling to the right like its components, whose amplitude is proportional to the cosine of <math>\varphi/2</math>.
* ''Constructive interference'': If the phase difference is an even multiple of {{pi}}: <math>\varphi = \ldots,-4\pi, -2\pi, 0, 2\pi, 4\pi,\ldots</math> then <math>\left|\cos(\varphi/2)\right| = 1</math>, so the sum of the two waves is a wave with twice the amplitude <math display="block">W_1 + W_2 = 2A\cos(kx - \omega t)</math>
* ''Destructive interference'': If the phase difference is an odd multiple of {{pi}}: <math>\varphi = \ldots,-3\pi,\, -\pi,\, \pi,\, 3\pi,\, 5\pi,\ldots</math> then <math>\cos(\varphi/2) = 0\,</math>, so the sum of the two waves is zero <math display="block">W_1 + W_2 = 0</math>

=== Between two plane waves ===
[[File:interference of plane waves 3.svg|class=skin-invert-image|thumb|Geometrical arrangement for two plane wave interference]] 
[[File:Interferences plane waves.jpg|thumb|Interference fringes in overlapping plane waves]]
A simple form of interference pattern is obtained if two [[plane wave]]s of the same frequency intersect at an angle.
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

<math display="block"> \Delta \varphi = \frac {2 \pi d} {\lambda} = \frac {2 \pi x \sin \theta} {\lambda}.</math>

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

<math display="block"> \frac {x \sin \theta} {\lambda} = 0, \pm 1, \pm 2, \ldots ,</math>

and are half a cycle out of phase when

<math display="block"> \frac {x \sin \theta} {\lambda} = \pm \frac {1}{2}, \pm \frac {3}{2}, \ldots </math>

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

<math display="block"> d_f = \frac {\lambda} {\sin \theta}</math>

and {{math|''d<sub>f</sub>''}} is known as the fringe spacing.  The fringe spacing increases with increase in [[wavelength]], and with decreasing angle {{math|θ}}.

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

=== Between two spherical waves ===
[[File:wavepanel.png|thumb|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, [[three-phase electric power|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 [[phasor]]s, each wave can be represented as <math>A e^{i \varphi_n}</math> for <math>N</math> waves from <math>n=0</math> to <math>n = N-1</math>, where
<math display="block">\varphi_n - \varphi_{n-1} = \frac{2\pi}{N}.</math>

To show that
<math display="block">\sum_{n=0}^{N-1} A e^{i \varphi_n} = 0</math>

one merely assumes the converse, then multiplies both sides by <math> e^{i \frac{2\pi}{N}}.</math>

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.