Coherence (physics)

[2] The amount of coherence can readily be measured by the interference visibility, which looks at the size of the interference fringes relative to the input waves (as the phase offset is varied); a precise mathematical definition of the degree of coherence is given by means of correlation functions.

Spatial coherence describes the correlation (or predictable relationship) between waves at different points in space, either lateral or longitudinal.

[5] Temporal coherence describes the correlation between waves observed at different moments in time.

In both cases, the fringe amplitude slowly disappears, as the path difference increases past the coherence length.

Coherence was originally conceived in connection with Thomas Young's double-slit experiment in optics but is now used in any field that involves waves, such as acoustics, electrical engineering, neuroscience, and quantum mechanics.

The property of coherence is the basis for commercial applications such as holography, the Sagnac gyroscope, radio antenna arrays, optical coherence tomography and telescope interferometers (Astronomical optical interferometers and radio telescopes).

As an example, consider two waves perfectly correlated for all times (by using a monochromatic light source).

[11]: 545-550 These states are unified by the fact that their behavior is described by a wave equation or some generalization thereof.

However, in optics one cannot measure the electric field directly as it oscillates much faster than any detector's time resolution.

is):[11]: 358–359, 560 Formally, this follows from the convolution theorem in mathematics, which relates the Fourier transform of the power spectrum (the intensity of each frequency) to its autocorrelation.

[11]: 572 Narrow bandwidth lasers have long coherence lengths (up to hundreds of meters).

The resulting visibility of the interference pattern (e.g. see Figure 4) gives the temporal coherence at delay

, an infinitely fast detector would measure an intensity that fluctuates significantly over a time t equal to

In some systems, such as water waves or optics, wave-like states can extend over one or two dimensions.

The range of separation between the two points over which there is significant interference defines the diameter of the coherence area,

Different points in the filament emit light independently and have no fixed phase-relationship.

Spatial coherence of laser beams also manifests itself as speckle patterns and diffraction fringes seen at the edges of shadow.

Its inventor, Dennis Gabor, produced successful holograms more than ten years before lasers were invented.

In February 2011 it was reported that helium atoms, cooled to near absolute zero / Bose–Einstein condensate state, can be made to flow and behave as a coherent beam as occurs in a laser.

[17][18] In the guided systems, the partial and full incoherence could be studied in terms of Gaussian shell model.

An absorbing polarizer rotated to any angle will always transmit half the incident intensity when averaged over time.

[22] Each slit acts as a separate but in-phase beam contributing to the intensity pattern on a screen.

[23]: 1057 As with light, transverse coherence (across the direction of propagation) of matter waves is controlled by collimation.

Because light, at all frequencies, travels the same velocity, longitudinal and temporal coherence are linked; in matter waves these are independent.

[22]: 154 The discovery of the Hanbury Brown and Twiss effect – correlation of light upon coincidence – triggered Glauber's creation[24] of uniquely quantum coherence analysis.

For instance, the laser, superconductivity and superfluidity are examples of highly coherent quantum systems whose effects are evident at the macroscopic scale.

For bosons, a Bose–Einstein condensate is an example of a system exhibiting macroscopic quantum coherence through a multiple occupied single-particle state.

Recently M. B. Plenio and co-workers constructed an operational formulation of quantum coherence as a resource theory.

In quantum mechanics for example one considers a probability field, which is related to the wave function

Low coherence can be caused by poor signal to noise ratio, and/or inadequate frequency resolution.

Two slits illuminated by one source show an interference pattern. The source is far to the left in the diagram, behind collimators that create a parallel beam. This combination ensures that a wave from the source strikes both slits at the same part of the wave cycle: the wave will have coherence .
Figure 1: The amplitude of a single frequency wave as a function of time t (red) and a copy of the same wave delayed by (blue). The coherence time of the wave is infinite since it is perfectly correlated with itself for all delays . [ 13 ] : 118
Figure 2: The amplitude of a wave whose phase drifts significantly in time as a function of time t (red) and a copy of the same wave delayed by (green). [ 14 ] At any particular time t the wave can interfere perfectly with its delayed copy. But, since half the time the red and green waves are in phase and half the time out of phase, when averaged over t any interference disappears at this delay.
Figure 3: The amplitude of a wavepacket whose amplitude changes significantly in time (red) and a copy of the same wave delayed by (green) plotted as a function of time t . At any particular time the red and green waves are uncorrelated; one oscillates while the other is constant and so there will be no interference at this delay. Another way of looking at this is the wavepackets are not overlapped in time and so at any particular time there is only one nonzero field so no interference can occur.
Figure 4: The time-averaged intensity (blue) detected at the output of an interferometer plotted as a function of delay τ for the example waves in Figures 2 and 3. As the delay is changed by half a period, the interference switches between constructive and destructive. The black lines indicate the interference envelope, which gives the degree of coherence . Although the waves in Figures 2 and 3 have different time durations, they have the same coherence time.
Figure 10: Waves of different frequencies interfere to form a localized pulse if they are coherent.
Figure 11: Spectrally incoherent light interferes to form continuous light with a randomly varying phase and amplitude