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Coherence (physics)

Coherence describes the fixed phase relationship and correlation between waves or quantum states. It distinguishes perfect, partial, temporal and spatial coherence and underpins interferometry, lasers, holography and decoherence.

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Overview

In physics, coherence refers to the degree to which oscillating quantities—most often electromagnetic waves—maintain a predictable phase relationship and correlation in time and space. When two wave sources share a constant phase difference, identical frequency and the same waveform, they are said to be perfectly coherent; this ideal situation produces stationary interference patterns. Practical sources are usually only partially coherent, but the concept of coherence remains central to understanding interference, diffraction and many measurement techniques. For more advanced treatments see advanced coherence theory.

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Interference finite coherence
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Key characteristics

Several distinct but related aspects of coherence are commonly considered:

  • Temporal coherence — how well a wave maintains a fixed phase relationship over time. It is related to the source bandwidth and determines the length of path difference over which stable interference occurs.
  • Spatial coherence — how uniform the phase is across different points of a wavefront; it controls visibility of interference across separated apertures.
  • Degree of coherence — a normalized measure of correlation between fields at two points; mathematically expressed by correlation functions and spectra.
  • Perfect vs partial coherence — perfect coherence is an idealization rarely reached in practice; most physical light lies between perfect coherence and complete incoherence.

Physical description and measures

Coherence can be characterized by correlation functions such as the mutual coherence function and the complex degree of coherence. These quantify how closely related two samples of a field are in phase and amplitude. The idea depends on phase stability (phase difference), frequency content (frequency) and waveform shape (waveform). Idealized descriptions treat coherence as stationary in time and space, an assumption that simplifies the analysis but does not hold for every real source. Coherence is a property of waves in general, whether classical waves or quantum probability amplitudes.

History and theoretical development

Interference experiments such as Young’s double-slit and Michelson interferometer exposed the practical importance of phase relations long before a formal theory was developed. In the 20th century, optical coherence theory matured to link statistical descriptions of light with observable interference. The concept was later generalized in quantum optics and quantum mechanics, where coherence describes superposition between states and is quantified by off-diagonal elements of a density matrix. For historical and mathematical context see foundational discussions.

Applications and examples

Coherence underlies many technologies and measurements. Examples include:

  • Interferometry for precision distance and surface measurements, including astronomy and metrology.
  • Lasers, which are highly temporally and spatially coherent compared with thermal sources, enabling tight focusing and long-range interference.
  • Holography and coherent imaging methods that rely on stable phase relationships to reconstruct wavefronts.
  • Spectroscopy and telecommunications, where coherence bandwidth and coherence time limit resolution and data-carrying capacity.

Distinctions and notable facts

Coherence is not the same as monochromaticity, polarization, or intensity. A source can be monochromatic but spatially incoherent, or polarized but temporally incoherent. In quantum contexts, loss of coherence—decoherence—explains how quantum superpositions become classical mixtures through interactions with environments; this is central to quantum computing and foundational studies. Practical analysis often treats coherence statistically via correlation functions that quantify how predictable a field is across space and time (correlation measures).

More description

All physical waves such as light waves, radar waves, sound waves or water waves can be coherent to other waves in a certain way, or there can be coherence between corresponding partial waves. The cause of the coherence may be a common history of generation of the waves. For example, if the same causal mechanism was at the root of the wave generation, there may be consistent oscillation patterns in the wave train that can be visualized later when partial waves are compared. If the wave amplitudes of two waves are directly correlated with each other, this can be seen in the appearance of stationary (spatially and temporally invariant) interference phenomena when the waves are superimposed. In other cases, a technically higher effort or a more complicated mathematical observation of the wave course is necessary to prove a coherence in the waves.

In simple cases, such as periodic waves, two partial waves are coherent if there is a fixed phase relationship between them. In optics, this phase relationship often means a constant difference between the phases of the oscillation period. Partial waves that overlap at a fixed location to a certain (time-averaged) intensity (for example, on an observation screen) can then, depending on the phase relationship, either amplify or cancel each other (complete coherence), amplify or attenuate each other a little (partial coherence), or cancel each other out to an average intensity (incoherence). Incoherence is present here above all at different frequencies, if all phase differences occur with equal frequency and thus no constructive or destructive interference is possible.

On the other hand, waves with different frequencies can also show coherence to each other. Technically, this type of coherence plays a role in frequency combing or in radar technology. This coherence is generated by mode coupling or frequency doubling or multiplication.

In wave fields, one can also distinguish the cases of a temporal and a spatial coherence, although normally both forms of coherence must be present. Temporal coherence exists when there is a fixed phase difference along the time axis (often figuratively equated with the spatial axis parallel to the direction of propagation). Spatial coherence exists when there is a fixed phase difference along a spatial axis (often reduced to the spatial axes perpendicular to the direction of propagation).

Mathematical representation

Coherence and correlation

The coherence required for interference capability in waves can be quantified using the correlation function. This function provides a measure of the similarity in time between two associated wave amplitudes.

The function

\Gamma_{\mathrm A\mathrm B}(\tau) = \langle E(\mathbf r_{\mathrm A},t)E^*(\mathbf r_{\mathrm B},t+\tau) \rangle = \lim \limits_{T \to \infty} \frac{1}{T} \int \limits_{-T/2}^{+T/2} {E(\mathbf r_{\mathrm A},t) E^*(\mathbf r_{\mathrm B},t+\tau) \textrm{d}t}

first defines the (complex) cross-correlation function between the time courses of two amplitudes under consideration. The two amplitudes are picked out at the location points A and \tau B of the wave Eand at a time difference of τ and tcompared as a function of time

The contrast function for spatiotemporal coherence provided by

K_{\mathrm A\mathrm B}(\tau) = \left| \frac{2\Gamma_{\mathrm A\mathrm B}(\tau)}{\Gamma_{\mathrm A\mathrm A}(0)+\Gamma_{\mathrm B\mathrm B}(0)} \right|

is given, now directly provides the strength of the coherence as a value between 0 and 1. In general, one distinguishes three cases:

K_{\mathrm A\mathrm B}(\tau)

= 1

 full coherence

0 <

K_{\mathrm A\mathrm B}(\tau)

< 1

 partial coherence

K_{\mathrm A\mathrm B}(\tau)

= 0

 complete incoherence

In the case of purely temporal coherence, only correlations with A = B are considered. Here the contrast function for temporal coherence provides

K(\tau) = \left| \frac{\Gamma(\tau)}{\Gamma(0)} \right|

the strength of the temporal coherence as a function of the time interval τ \tau . K(\tau)has maximum value 1 at τ \tau=0 and decreases to 0 more or less rapidly depending on the coherence. The coherence time τ \tau_c is defined as the time interval τ \tau , at which the contrast function has dropped to 1/e. If the coherence between different waves is to be calculated, the cross-correlation function

\Gamma(\tau)=\langle E_1(t)E_2^*(t+\tau) \rangle

of the waves E_{1}and E_{2}used.

In the case of pure spatial coherence, only correlations with τ are \tau=0considered. Here the contrast function for spatial coherence yields

K_{\mathrm A\mathrm B} = \left| \frac{2\Gamma_{\mathrm A\mathrm B}}{\Gamma_{\mathrm A\mathrm A}+\Gamma_{\mathrm B\mathrm B}} \right|

the strength of spatial coherence between points A and B. A volume in which all pairs of points A, B have contrast K_{\mathrm A\mathrm B} > 1/e , forms a so-called coherence volume within which spatial coherence exists. Usually the term spatial coherence is understood to mean only coherence transverse to the direction of propagation of the wave, which should more precisely be called transverse spatial coherence. The spatial coherence along the direction of propagation, i.e. the longitudinal spatial coherence, is often equated with the temporal coherence, which is only approximately correct.

Multibeam interference

The mathematical definition of coherence shown only describes the correlation between two points of a wave. In many applications, however, the condition must be fulfilled that a very large number of partial waves can be superimposed to form a common interference pattern. In this case, the pairwise coherence of the partial waves alone is not sufficient. The concept of coherence must be extended for this purpose or linked with additional conditions.

In the example of a diffraction grating in optics, for example, where a very large number of partial waves must interfere, spatial coherence in pairs is not yet sufficient to make sharp diffraction spectra visible. In addition, a simultaneous correlation between the phases of all partial waves must be present so that the partial beams capable of interference in pairs come to coincide in a common diffraction maximum on the screen. This condition is especially fulfilled when plane wavefronts meet a plane diffraction grating. Two further applications in which many-beam interference plays a role are Bragg reflection and the Fabry-Pérot interferometer.

Questions and answers

Q: What is coherence in advanced physics?

A: Coherence in advanced physics is a phenomenon of electromagnetic waves.

Q: When are two wave sources perfectly coherent?

A: Two wave sources are perfectly coherent if they have a constant phase difference and the same frequency, and the same waveform.

Q: What happens when two wave sources are perfectly coherent?

A: When two wave sources are perfectly coherent, the waves plot as identical: their peaks and troughs occur at the same time, and they have the same amplitude.

Q: What does coherence produce?

A: Coherence produces stationary (i.e. temporally and spatially constant) interference.

Q: Is coherence an ideal property of waves?

A: Yes, coherence is an ideal property of waves.

Q: What does coherence describe?

A: More generally, coherence describes all properties of the correlation between physical quantities of a single wave, or between several waves or wave packets.

Q: Why has coherence become an important concept in quantum physics?

A: Coherence has become an important concept in quantum physics as it allows an understanding of the physics of waves.

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Coherence Interferometry

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