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# interference

[in-ter-feer-uhns]
interference, in physics, the effect produced by the combination or superposition of two systems of waves, in which these waves reinforce, neutralize, or in other ways interfere with each other. Interference is observed in both sound waves and electromagnetic waves, especially those of visible light and radio.

## Interference in Sound Waves

When two sound waves occur at the same time and are in the same phase, i.e., when the condensations of the two coincide and hence their rarefactions also, the waves reinforce each other and the sound becomes louder. This is known as constructive interference. On the other hand, two sound waves occurring simultaneously and having the same intensity neutralize each other if the rarefactions of the one coincide with the condensations of the other, i.e., if they are of opposite phase. This canceling is known as destructive interference. In this case, the result is silence.

Alternate reinforcement and neutralization (or weakening) take place when two sound waves differing slightly in frequency are superimposed. The audible result is a series of pulsations or, as these pulsations are called commonly, beats, caused by the alternate coincidence of first a condensation of the one wave with a condensation of the other and then a condensation with a rarefaction. The beat frequency is equal to the difference between the frequencies of the interfering sound waves.

## Interference in Light Waves

Light waves reinforce or neutralize each other in very much the same way as sound waves. If, for example, two light waves each of one color (monochromatic waves), of the same amplitude, and of the same frequency are combined, the interference they exhibit is characterized by so-called fringes—a series of light bands (resulting from reinforcement) alternating with dark bands (caused by neutralization). Such a pattern is formed either by light passing through two narrow slits and being diffracted (see diffraction), or by light passing through a single slit. In the case of two slits, each slit acts as a light source, producing two sets of waves that may combine or cancel depending upon their phase relationship. In the case of a single slit, each point within the slit acts as a light source. In all cases, for light waves to demonstrate such behavior, they must emanate from the same source; light from distinct sources has too many random differences to permit interference patterns.

The relative positions of light and dark lines depend upon the wavelength of the light, among other factors. Thus, if white light, which is made up of all colors, is used instead of monochromatic light, bands of color are formed because each color, or wavelength, is reinforced at a different position. This fact is utilized in the diffraction grating, which forms a spectrum by diffraction and interference of a beam of light incident on it. Newton's rings also are the result of the interference of light. They are formed concentrically around the point of contact between a glass plate and a slightly convex lens set upon it or between two lenses pressed together; they consist of bright rings separated by dark ones when monochromatic light is used, or of alternate spectrum-colored and black rings when white light is used. Various natural phenomena are the result of interference, e.g., the colors appearing in soap bubbles and the iridescence of mother-of-pearl and other substances.

## Interference as a Scientific Tool

The experiments of Thomas Young first illustrated interference and definitely pointed the way to a wave theory of light. A. J. Fresnel's experiments clearly demonstrated that the interference phenomena could be explained adequately only upon the basis of a wave theory. The thickness of a very thin film such as the soap-bubble wall can be measured by an instrument called the interferometer. When the wavelength of the light is known, the interferometer indicates the thickness of the film by the interference patterns it forms. The reverse process, i.e., the measurement of the length of an unknown light wave, can also be carried out by the interferometer.

The Michelson interferometer used in the Michelson-Morley experiment of 1887 to determine the velocity of light had a half-silvered mirror to split an incident beam of light into two parts at right angles to one another. The two halves of the beam were then reflected off mirrors and rejoined. Any difference in the speed of light along the paths could be detected by the interference pattern. The failure of the experiment to detect any such difference threw doubt on the existence of the ether and thus paved the way for the special theory of relativity.

Another type of interferometer devised by Michelson has been applied in measuring the diameters of certain stars. The radio interferometer consists of two or more radio telescopes separated by fairly large distances (necessary because radio waves are much longer than light waves) and is used to pinpoint and study various celestial sources of radiation in the radio range. Astronomical interferometers consisting of two or more optical telescopes are used to enhance visible images of distant celestial objects. See radio astronomy; virtual telescope.

In physics, the net effect of combining two or more wave trains moving on intersecting or coincident paths. Constructive interference occurs if two components have the same frequency and phase; the wave amplitudes are reinforced. Destructive interference occurs when the two waves are out of phase by one-half period (see periodic motion); if the waves are of equal amplitude, they cancel each other. Two waves moving in the same direction but having slightly different frequencies interfere constructively at regular intervals, resulting in a pulsating frequency called a beat. Two waves traveling in opposite directions but having equal frequencies interfere constructively in some places and destructively in others, resulting in a standing wave.

In physics, interference is the addition (superposition) of two or more waves that result in a new wave pattern.

As most commonly used, the term interference usually refers to the interaction of waves which 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.

Two non-monochromatic waves are only fully coherent with each other if they both have exactly the same range of wavelengths and the same phase differences at each of the constituent wavelengths.

The total phase difference is derived from the sum of both the path difference and the initial phase difference (if the waves are generated from 2 or more different sources). It can then be concluded whether the waves reaching a point are in phase (constructive interference) or out of phase (destructive interference).

## Theory

The principle of superposition of waves states that the resultant displacement at a point is equal to the vector sum of the displacements of different waves at that point. If a crest of a wave meets a crest of another wave at the same point then the crests interfere constructively and the resultant wave amplitude is greater. If a crest of a wave meets a trough of another wave then they interfere destructively, and the overall amplitude is decreased.

This form of interference can occur whenever a wave can propagate from a source to a destination by two or more paths of different length. Two or more sources can only be used to produce interference when there is a fixed phase relation between them, but in this case the interference generated is the same as with a single source; see Huygens' principle.

## Experiments

Thomas Young's double-slit experiment showed interference phenomena where two beams of light which are coherent interfere to produce a pattern.

The beams of light both have the same wavelength range and at the center of the interference pattern. They have the same phases at each wavelength, as they both come from the same source.

## Interference patterns

For two coherent sources, the spatial separation between sources is half the wavelength times the number of nodal lines.

Light from any source can be used to obtain interference patterns, for example, Newton's rings can be produced with sunlight. However, in general white light is less suited for producing clear interference patterns, as it is a mix of a full spectrum of colours, that each have different spacing of the interference fringes. Sodium light is close to monochromatic and is thus more suitable for producing interference patterns. The most suitable is laser light because it is almost perfectly monochromatic.

## Constructive and destructive interference

Consider two waves that are in phase,with amplitudes A1 and A2. Their troughs and peaks line up and the resultant wave will have amplitude A = A1 + A2. This is known as constructive interference.

If the two waves are π radians, or 180°, out of phase, then one wave's crests will coincide with another wave's troughs and so will tend to cancel out. The resultant amplitude is A = |A1A2|. If A1 = A2, the resultant amplitude will be zero. This is known as destructive interference.

When two sinusoidal waves superimpose, the resulting waveform depends on the frequency (or wavelength) amplitude and relative phase of the two waves. If the two waves have the same amplitude A and wavelength the resultant waveform will have an amplitude between 0 and 2A depending on whether the two waves are in phase or out of phase.

combined
waveform
|- wave 1
wave 2

Two waves in phase Two waves 180° out
of phase

## General quantum interference

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

$|psi rang = sum_i |irang psi_i$

where the $|irang$s specify the different quantum "alternatives" available (technically, they form an eigenvector basis) and the $psi_i$ are the probability amplitude coefficients, which are complex numbers.

The probability of observing the system making a transition or quantum leap from state $Psi$ to a new state $Phi$ is the square of the modulus of the scalar or inner product of the two states:

$prob\left(psi Rightarrow phi\right) = |lang psi |phi rang|^2 = |sum_ipsi^*_i phi_i |^2 =$ $= sum_\left\{ij\right\} psi^*_i psi_j phi^*_jphi_i= sum_\left\{i\right\} |psi_i|^2|phi_i|^2 + sum_\left\{ij;i ne j\right\} psi^*_i psi_j phi^*_jphi_i$

where $psi_i = lang i|psi rang$ (as defined above) and similarly $phi_i = lang i|phi rang$ are the coefficients of the final state of the system. * is the complex conjugate so that $psi_i^* = lang psi|i rang$, etc.

Now let's consider the situation classically and imagine that the system transited from $|psi rang$ to $|phi rang$ via an intermediate state $|irang$. 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

$prob\left(psi Rightarrow phi\right) = sum_i prob\left(psi Rightarrow i Rightarrow phi\right) =$ $= sum_i |lang psi |i rang|^2|lang i|phi rang|^2 = sum_i|psi_i|^2 |phi_i|^2.$

The classical and quantum derivations for the transition probability differ by the presence, in the quantum case, of the extra terms $sum_\left\{ij;i ne j\right\} psi^*_i psi_j phi^*_jphi_i$; these extra quantum terms represent interference between the different $i ne j$ 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 $|irang$ is measured or coupled with the environment.

## Examples

A conceptually simple case of interference is a small (compared to wavelength) source - say, a small array of regularly spaced small sources (see diffraction grating).

Consider the case of a flat boundary (say, between two media with different densities or simply a flat mirror), onto which the plane wave is incident at some angle. In this case of continuous distribution of sources, constructive interference will only be in specular direction - the direction at which angle with the normal is exactly the same as the angle of incidence. Thus, this results in the law of reflection which is simply the result of constructive interference of a plane wave on a plane surface.