Wave interference

In physics, interference is a phenomenon in which two coherent waves are combined by adding their intensities or displacements with due consideration for their phase difference. The resultant wave may have greater amplitude or lower amplitude if the two waves are in phase or out of phase, respectively.
Interference effects can be observed with all types of waves, for example, light, radio, acoustic, surface water waves, gravity waves, or matter waves as well as in loudspeakers as electrical waves.

Etymology

The word interference is derived from the Latin words inter which means "between" and fere which means "hit or strike", and was used in the context of wave superposition by Thomas Young in 1801.

Mechanisms

The 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 vector sum of the amplitudes 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 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 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.
Constructive interference occurs when the phase difference between the waves is an even multiple of , whereas destructive interference occurs when the difference is an odd multiple of. 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 coatings and interferometers.
In addition to the 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 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 its components, whose amplitude is proportional to the cosine of.

Constructive interference: If the phase difference is an even multiple of : 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 : then, so the sum of the two waves is zero
Between two plane waves

A simple form of interference pattern is obtained if two plane waves 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
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 is known as the fringe spacing. The fringe spacing increases with increase in wavelength, and with decreasing angle.
The fringes are observed wherever the two waves overlap and the fringe spacing is uniform throughout.

Between two spherical waves

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. 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 wave interference

Because the frequency of light waves 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 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 is:
where represents the magnitude of the displacement, represents the phase and represents the angular frequency.
The displacement of the summed waves is
The intensity of the light at 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 2. 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 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.
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, 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. 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.
This has also been observed for widefield interference between two incoherent laser sources.
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.