# coherence

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## coherence,

constant phase difference in two or more WavesWaves
(Women Appointed for Voluntary Emergency Service), U.S. navy organization, created (1942) in World War II to release male naval personnel for sea duty. The organization was commanded until 1946 by Mildred Helen McAfee.
over time. Two waves are said to be in phase if their crests and troughs meet at the same place at the same time, and the waves are out of phase if the crests of one meet the troughs of another. The waves are incoherent if the crests and troughs meet randomly. Coherence underlies a variety of physical phenomena, such as interferenceinterference,
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.
and diffractiondiffraction,
bending of waves around the edge of an obstacle. When light strikes an opaque body, for instance, a shadow forms on the side of the body that is shielded from the light source.
. Coherence is also responsible for many of the remarkable properties of laserlaser
[acronym for light amplification by stimulated emission of radiation], device for the creation, amplification, and transmission of a narrow, intense beam of coherent light. The laser is sometimes referred to as an optical maser.
radiation; laser light is coherent, which is to say that the light waves from a laser are all in phase.

## Coherence

The attribute of two or more waves, or parts of a wave, whose relative phase is constant during the resolving time of the observer. The concept has been developed most extensively in optics, but is applicable to all wave phenomena.

Consider two waves, with the same mean angular frequency ω, given by Eqs. (1)

(1)
(2)
and (2). It is convenient, and no restriction, to choose both A and B real. These expressions as they stand could describe de Broglie waves in quantum mechanics. For real waves, such as components of the electric field in light or radio beams, or the pressure oscillations in sound, it is necessary to retain only the real parts of these and subsequent expressions. The frequency spectrum is assumed to be narrow, in the sense that a Fourier analysis of expressions (1) and (2) gives appreciable contributions only for angular frequencies close to &ohgr;. This assumption means that, on the average, δA(t) and δB(t) do not change much per period. See Electromagnetic radiation, Quantum mechanics, Sound

Suppose that the waves are detected by an apparatus with resolving time T, that is, T is the shortest interval between two events for which the events do not seem to be simultaneous. For the human eye and ear, T is about 0.1 s, while a fast electronic device might have a T of 10-10 s. If the relative phase δ(t), given by Eq. (3),

(3)
does not, on the average, change noticeably during T, then the waves are coherent. If during T there are sufficient random fluctuations for all values of δ(t), modulus 2&pgr;, to be equally probable, then the waves are incoherent. If during T there are noticeable random fluctuations in δ(t), but not enough to make the waves completely incoherent, then the waves are partially coherent. These distinctions are not useful unless T is specified. On the one hand, only waves that have existed forever and that fill all of space can have absolutely fixed frequency and phase. On the other hand, two independent sound waves in the phases change appreciably in 0.01 s would seem incoherent to the human ear, but would seem highly coherent to a fast electronic device.

The degree of coherence is related to the interference patterns that can be observed when the two beams are combined. See Interference of waves

Coherence is also used to describe relations between phases within the same beam. Suppose that a wave represented by Eq. (1) is passing a fixed observer characterized by a resolving time T. The phase δA may fluctuate, perhaps because the source of the wave contains many independent radiators. The coherence time ΔtW of the wave is defined to be the average time required for δA(t) to fluctuate appreciably at the position of the observer. If ΔtW is much greater than T, the wave is coherent; if ΔtW is of the order of T, the wave is partially coherent; and if ΔtW is much less than T, the wave is incoherent. These concepts are very close to those developed above.

Extended sources give partial coherence and produce interference fringes with visibility V less than unity. A. A. Michelson exploited this fact with his stellar interferometer, a modified double-slit arrangement with movable mirrors that permit adjustment of the effective separation D of the slits. It can be shown that if the source is a uniform disk of angular diameter Θ, then the smallest value of D that gives zero V is 1.22λ/Θ. The same approach has also been applied in radio astronomy. A different technique, developed by R. Hanbury Brown and R. Q. Twiss, measures the correlation between the intensifies received by separated detectors with fast electronics. See Interferometry

Because they are highly coherent sources, lasers and masers provide very large intensities per unit frequency. See Laser

Photon statistics is concerned with the probability distribution describing the number of photons incident on a detector, or present in a cavity. By extension, it deals with the correlation properties of beams of light.

According to the quantum theory of electromagnetism, quantum electrodynamics, light is made up of particles called photons, each of which possesses an energy E of &planck;&ohgr;, where &planck; is Planck's constant divided by 2&pgr; and &ohgr; is the angular frequency of the light (the frequency multiplied by 2&pgr;). In general, however, the photon number is an intrinsically uncertain quantity. It is impossible to precisely specify both the phase &phgr; = &ohgr;t of a wave and the number of photons n &ap; E/(&planck;&ohgr;) that it contains; the uncertainties of these two conjugate variables must satisfy ΔnΔ&phgr; ≥ &half;. For a beam to be coherent in the sense of having a well-defined phase, it must not be describable in terms of a fixed number of particles. (Lacking a fixed phase, a single photon may interfere only with itself, not with other photons.) See Photon, Quantum electrodynamics

The most familiar example of this uncertainty is shot noise, the randomness of the arrival times of individual photons. There is no correlation between photons in the coherent state emitted by a classical source such as an ideal laser or radio transmitter, so the number of photons detected obeys Poisson statistics, displaying an uncertainty equal to the square root of the mean. The shot noise constitutes the dominant source of noise at low light levels, and may become an important factor in optical communications as well as in high-precision optical devices (notably those that search for gravitational radiation).

## coherence

(koh-heer -ĕns) The degree to which an oscillating quantity maintains a constant phase and amplitude relationship at points displaced in space or time. Hence the coherence width of a train of waves describes the distance along a wavefront over which the oscillations are appreciably correlated while the coherence time defines the time for which the character of the wavetrain remains more or less unchanged. A radio interferometer measures coherence width. This is related to the shape of the radio source and can be used to generate an image of it. See also autocorrelation function.

## Coherence

the coordinated course of several oscillatory or wave processes with respect to time that is manifested upon addition (interference). Oscillations are said to be coherent if their phase difference remains constant over time and, when the oscillations are added, defines the amplitude of the resultant oscillation. Two harmonic (sinusoidal) oscillations of a given frequency are always coherent. A harmonic oscillation is described by the expression

(1) x = A cos (2πvt + ϕ)

where x is the oscillating quantity (for example, the displacement of a pendulum from a state of equilibrium; the intensity of electric and magnetic fields). The frequency of a harmonic oscillation and its amplitude A and phase ϕ are constant with respect to time. When two harmonic oscillations of identical frequency v but with different amplitudes A1 and A2 and phases ϕ1 and ϕ2 are added, a harmonic oscillation of the same frequency is formed. The amplitude of the resultant oscillation

may vary from A1 + A2 to A1 – A2, depending on the phase difference ϕ1 – ϕ2 (Figure 1). The intensity of the resultant oscillation, which is proportional to Ar2, also depends on the phase difference.

Figure 1. Addition of two harmonic oscillations (broken line) with amplitudes A1 and A2 for various phase differences. The resultant oscillation is shown by the solid line.

In reality, ideally harmonic oscillations cannot be produced, since the amplitude, frequency, and phase of oscillations in real oscillatory processes change continuously and randomly over time. The resultant amplitude Ar depends essentially on the rate of change of the phase difference. If the changes are so fast that they cannot be recorded by an instrument, it is possible only to measure the average amplitude of the resultant oscillation,Ā. Here, since the average value of cos (ϕ1 – ϕ2) is equal to 0, the average intensity of the resultant oscillation is equal to the sum_of the_average intensities of the initial oscillations, Ār2 = Ā12 + Ā22, and thus is independent of their phases. The initial oscillations are incoherent. The random_rapid changes in amplitude also disrupt the coherence (Ā1Ā2 = 0 ).

However, if the phases of the oscillations ϕ1 and ϕ2 change but their difference (ϕ1 – ϕ2 remains constant, the intensity of the resultant oscillation, as in the case of ideally harmonic oscillations, is defined by the difference in the phases of the added oscillations—that is, coherence occurs. If the phase difference of the two oscillations changes very slowly, then the oscillations are said to remain coherent for a certain period, until their phase difference changes by a quantity comparable to π.

The phases of a given oscillation can be compared at different instants t1 and t2, separated by the interval τ. If the nonharmonic nature of the oscillation is manifested in a random change of its phase over time, the change in the phase of the oscillation may exceed π for sufficiently large τ. This means that after some interval r the harmonic oscillation “forgets” its original phase and becomes incoherent “with respect to itself.” The time τ is called the coherence time of a nonharmonic oscillation, or the period of the harmonic train. As one harmonic train expires, it is replaced by another of the same frequency but different phase.

During propagation of a plane, monochromatic electromagnetic wave in a homogeneous medium, the intensity of the electric field E in the direction of propagation of the wave ox at time t is equal to

where λ = cT is the wavelength, c is the rate of propagation of the wave, and T is the oscillation period. The phase of the oscillations at some specific point in space is retained only for the coherence time τ. In this period the wave covers a distance and the oscillation of E at points that are a distance of away from each other is found to be incoherent in the direction of wave propagation. The distance, equal to in the direction of propagation of the plane wave, at which the random changes in the phase of the oscillations reach a quantity comparable to π, is called the coherence length, or the train length.

Visible sunlight, which occupies the band from 4 × 1014 to 8 × 1014 hertz on the scale of electromagnetic frequencies, may be considered a harmonic wave with rapidly changing amplitude, frequency, and phase. Here the train length is approximately 10-4 cm. The light radiated by a rarefied gas in the form of narrow spectral lines is closer to monochromatic light. The phase of such light is virtually unchanged at a distance of 10 cm. The train length of laser radiation may be several kilometers or more. In the radio band more monochromatic sources of oscillations exist, and the wavelength λ is many times greater than for visible light. The train length of radio waves may greatly exceed the dimensions of the solar system.

All of the above is true for a plane wave. However, an ideally plane wave is just as unrealizable as an ideally harmonic oscillation. In real wave processes the amplitude and phase of oscillations change not only in the direction of propagation but also in a plane perpendicular to it. Random changes in the phase difference at two points that lie in this plane increase with the distance between them. The coherence of oscillations is reduced at these points and disappears at some distance l, when the random changes in phase difference become comparable to π. The term “spatial coherence,” in contrast to time coherence, which is associated with the degree of monochromaticity of a wave, is used to describe the coherent properties of a wave in a plane perpendicular to the direction of its propagation. The entire space occupied by a wave may be divided into regions, in each of which the wave maintains its coherence. The volume of such a region (the coherence volume) is approximately equal to the product of the train length and the area of a circle of diameter l (the spatial coherence length).

The disruption of spatial coherence is associated with the peculiarities of the processes of wave radiation and formation. For example, the spatial coherence of a light wave radiated by an extended hot body vanishes at a distance of just a few wavelengths from its surface, since different parts of the hot body radiate independently of each other. As a result, instead of one plane wave the source radiates a set of plane waves that propagate in all possible directions. As the distance from the heat source (of finite dimensions) increases, the wave becomes more nearly a plane wave. The spatial coherence length l increases in proportion to λrR where R is the distance to the source and r is the dimensions of the source. This makes possible observation of the light interference of stars, even though they are heat sources of tremendous size. By measuring l for the light from the nearest stars, their dimensions r may be determined. The quantity λ/r is called the angle of coherence. As the distance from the source increases, the intensity of the light decreases as 1/R2.

Therefore the production of intensive radiation of high spatial coherence by a hot body is impossible.

The light wave radiated by a laser is formed as a result of coordinated stimulated emission of light throughout the entire active substance. Therefore the spatial coherence of light at the discharge opening of the laser is retained throughout the entire cross section of the beam. Laser radiation has tremendous spatial coherence (that is, high directionality as compared with the radiation of a hot body). By using a laser it is possible to produce light whose coherence volume exceeds by a factor of 1017 the coherence volume of a light wave of the same intensity produced from the most monochromatic nonlaser light sources.

In optics the most widely used method of producing two coherent waves is to split a wave radiated by a single nonmonochromatic source into two waves that follow different paths but ultimately meet at a point, where addition takes place (Figure 2). If the delay of one wave with respect to the other caused by the difference in the paths traversed is less than the train period, the oscillations will be coherent at the point of addition, and interference of light will be observed. When the difference in the paths of the two waves approximates the train length, the coherence of the beams is reduced. The variations in illumination over the screen are reduced, and the illumination I tends toward a constant value equal to the sum of the intensities of the two waves incident on the screen. In the case of a nonpoint (extended) heat source the two rays, upon reaching points A and B, may prove to be incoherent because of the spatial incoherence of the radiated wave. In this case interference is not observed, since the interference fringes from various points of the source are displaced with respect to each other by a distance greater than the width of the fringes.

Figure 2. A very simple device that makes possible the production of two coherent waves (an interferometer). The damper impedes direct passage of light from the source to the screen.

The concept of coherence, which originally arose in the classical theory of oscillations and waves, also is used with respect to the objects and processes described by quantum mechanics (such as atomic particles and solids).

### REFERENCES

Landsberg, G. S. Optika, 4th ed. Moscow, 1957.
Gorelik, G. S. Kolebaniia i volny, 2nd ed. Moscow, 1959.
Fabrikant, V. A. “Novoe o kogerentnosti.” Fizika v shkole, 1968, no. 1.
Françon, M., and S. Slanski. Kogerentnost’ v optike. Moscow, 1968. (Translated from French.)
Martinsen, V., and E. Shpiller. “Chto takoe kogerentnost’.” Priroda, 1968, no. 10.

A. V. FRANTSESSON

## coherence

[kō′hir·əns]
(physics)
The existence of a correlation between the phases of two or more waves, so that interference effects may be produced between them, or of a correlation between the phases of part of a single wave.
Property of moving in unison, such as is characteristic of the particles in a synchrotron.
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