Editing Wave interference (section)

=== Optical wave interference ===
[[File:Optical flat interference.svg|thumb|upright=1.55|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 (~10<sup>14</sup> Hz) is too high for currently available detectors to detect the variation of the electric field of the light, it is possible to observe only the [[intensity (physics)|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 {{math|'''r'''}} is:

<math display="block">U_1 (\mathbf r,t) = A_1(\mathbf r) e^{i [\varphi_1 (\mathbf r) - \omega t]}</math>
<math display="block">U_2 (\mathbf r,t) = A_2(\mathbf r) e^{i [\varphi_2 (\mathbf r) - \omega t]}</math>

where {{math|A}} represents the magnitude of the displacement, {{math|φ}} represents the phase and {{math|ω}} represents the [[angular frequency]].

The displacement of the summed waves is

<math display="block">U (\mathbf r,t) = A_1(\mathbf r) e^{i [\varphi_1 (\mathbf r) - \omega t]}+A_2(\mathbf r) e^{i [\varphi_2 (\mathbf r) - \omega t]}.</math>

The intensity of the light at {{math|'''r'''}} is given by

<math display="block"> I(\mathbf r) = \int U (\mathbf r,t) U^* (\mathbf r,t) \, dt \propto A_1^2 (\mathbf r)+ A_2^2 (\mathbf r) + 2 A_1 (\mathbf r) A_2 (\mathbf r) \cos [\varphi_1 (\mathbf r)-\varphi_2 (\mathbf r)]. </math>

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

<math display="block"> I(\mathbf r) =  I_1 (\mathbf r)+ I_2 (\mathbf r) + 2 \sqrt{ I_1 (\mathbf r) I_2 (\mathbf r)} \cos [\varphi_1 (\mathbf r)-\varphi_2 (\mathbf r)].</math>

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 2{{pi}}. 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.

Classically the two waves must have the same [[Polarization (waves)|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 (waves)#Polarization state|polarization state]].

Quantum mechanically the theories of Paul Dirac and Richard Feynman offer a more modern approach.  Dirac showed that every quanta or photon of light acts on its own which he famously stated as "every photon interferes with itself". Richard Feynman showed that by evaluating a path integral where all possible paths are considered, that a number of higher probability paths will emerge.  In thin films for example, film thickness which is not a multiple of light wavelength will not allow the quanta to traverse, only reflection is possible.  

==== 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 (but shorter than their coherence time), 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-vapor lamp|sodium-]] or [[mercury-vapor lamp]]s 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.<ref>{{cite book |first=W. H. |last=Steel |title=Interferometry |year=1986 |publisher=Cambridge University Press |location=Cambridge |isbn=0-521-31162-4 }}</ref> 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 thus 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.<ref>{{cite journal | last1 = Pfleegor | first1 = R. L. | last2 = Mandel | first2 = L. | year = 1967 | title = Interference of independent photon beams | doi = 10.1103/physrev.159.1084 | journal = Phys. Rev. | volume = 159 | issue = 5| pages = 1084–1088 | bibcode = 1967PhRv..159.1084P }}</ref>
This has also been observed for widefield interference between two incoherent laser sources.<ref>{{cite journal|last=Patel|first=R.|author2=Achamfuo-Yeboah, S. |author3=Light R.|author4=Clark M.|title= Widefield two laser interferometry|journal=Optics Express|date=2014|volume=22|issue=22|pages=27094–27101|url=https://www.osapublishing.org/oe/abstract.cfm?uri=oe-22-22-27094|bibcode=2014OExpr..2227094P|doi=10.1364/OE.22.027094|pmid=25401860|doi-access=free}}</ref>

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.<ref name="Born and Wolf">{{cite book |first1=Max |last1=Born |author-link=Max Born |first2=Emil |last2=Wolf |year=1999 |title=[[Principles of Optics]] |publisher=Cambridge University Press |location=Cambridge |isbn=0-521-64222-1 }}</ref>

==== Optical arrangements ====
To generate interference fringes, light from the source has to be divided into two waves which then have 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 experiment|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.

<gallery mode="packed" caption="Iridiscence caused by thin-film interference">
File:Samsung Galaxy A50 back 2.jpg|Smartphone with iridescent back panel
File:Dieselrainbow.jpg|An oil spill
File:Soap bubble sky.jpg|White light interference in a soap bubble.
</gallery>