Coherence

coherence, constant phase difference in two or more Waves 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 interference and diffraction. Coherence is also responsible for many of the remarkable properties of laser radiation; laser light is coherent, which is to say that the light waves from a laser are all in phase.

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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, δ_A(t) and δ_B(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 δ_A may fluctuate, perhaps because the source of the wave contains many independent radiators. The coherence time Δt_W of the wave is defined to be the average time required for δ_A(t) to fluctuate appreciably at the position of the observer. If Δt_W is much greater than T, the wave is coherent; if Δt_W is of the order of T, the wave is partially coherent; and if Δt_W 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 ℏ&ohgr;, where ℏ 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 ≈ E/(ℏ&ohgr;) that it contains; the uncertainties of these two conjugate variables must satisfy ΔnΔ&phgr; ≥ ½. 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).