Coherence (physics)

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In physics, coherence (from Latin: cohaerere = to hang together) refers to the property of an extended wave field that the instantaneous deflections at different locations change in the same way over time except for a phase shift that remains constant. As a result, spatial stationary interference can be seen when coherent waves are superimposed. The absence of coherence is called incoherence.

Derived from this, in quantum mechanics one speaks of coherent superposition of different states, if they have to be added like vectors under consideration of their quantum mechanical phases.

Characteristic for the coherence of two waves arriving at the same place is that their amplitudes add up. In the case of incoherence, their intensities, i.e. the (absolute) squares of their amplitudes, add up.

While the frequently chosen mathematical description of a wave as a sinusoid is thought to be unlimited in time and space, real physical waves are limited in time and space. Also, two waves generated by different arrangements usually have slightly different frequencies. Therefore, the presence of coherence usually indicates that the waves have a common or coherent history of origin. Depending on the duration of this emergence, the coherence can thus be limited in time. The path length covered in the process is called the coherence length, which measures the spatial extent of their coherence.

Coherence plays a role in all areas of physics where interference is observed, especially in laser optics, spectroscopy and interferometry. It does not matter for the significance of coherence whether light waves or matter waves are involved. Since it is possible, especially in laser technology, to generate numerous copies of individual photons with a coherent history of origin, coherence is also of great importance in their fields of application, such as the creation of holograms, quantum cryptography or signal processing.

The correlation integral serves as a measure of the interference capability of two waves and thus of the coherence of the two.

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 and at a time difference of τ and compared 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 $\tau=0$ considered. 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.