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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Coherence (physics) | 1/4 | https://en.wikipedia.org/wiki/Coherence_(physics) | reference | science, encyclopedia | 2026-05-05T13:41:37.176399+00:00 | kb-cron |
In physics, coherence expresses the potential for two waves to interfere. Two monochromatic beams from a single source always interfere. Even for wave sources that are not strictly monochromatic, they may still be partly coherent. When interfering, two waves add together to create a wave of greater amplitude than either one (constructive interference) or subtract from each other to create a wave of minima which may be zero (destructive interference), depending on their relative phase. Constructive or destructive interference are limit cases, and two waves always interfere, even if the result of the addition is complicated or not remarkable. Two waves with constant relative phase will be coherent. 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. More broadly, coherence describes the statistical similarity of a field, such as an electromagnetic field or quantum wave packet, at different points in space or time.
== Qualitative concept ==
Coherence controls the visibility or contrast of interference patterns. For example, visibility of the double slit experiment pattern requires that both slits be illuminated by a coherent wave as illustrated in the figure. Large sources without collimation or sources that mix many different frequencies will have lower visibility. Coherence contains several distinct concepts. Spatial coherence describes the correlation (or predictable relationship) between waves at different points in space, either lateral or longitudinal. Temporal coherence describes the correlation between waves observed at different moments in time. Both are observed in the Michelson–Morley experiment and Young's interference experiment. Once the fringes are obtained in the Michelson interferometer, when one of the mirrors is moved away gradually from the beam-splitter, the time for the beam to travel increases and the fringes become dull and finally disappear, showing temporal coherence. Similarly, in a double-slit experiment, if the space between the two slits is increased, the coherence dies gradually and finally the fringes disappear, showing spatial coherence. 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).
== Mathematical definition ==
The coherence function between two signals
x
(
t
)
{\displaystyle x(t)}
and
y
(
t
)
{\displaystyle y(t)}
is defined as
γ
x
y
2
(
f
)
=
|
S
x
y
(
f
)
|
2
S
x
x
(
f
)
S
y
y
(
f
)
{\displaystyle \gamma _{xy}^{2}(f)={\frac {|S_{xy}(f)|^{2}}{S_{xx}(f)S_{yy}(f)}}}
where
S
x
y
(
f
)
{\displaystyle S_{xy}(f)}
is the cross-spectral density of the signal and
S
x
x
(
f
)
{\displaystyle S_{xx}(f)}
and
S
y
y
(
f
)
{\displaystyle S_{yy}(f)}
are the power spectral density functions of
x
(
t
)
{\displaystyle x(t)}
and
y
(
t
)
{\displaystyle y(t)}
, respectively. The cross-spectral density and the power spectral density are defined as the Fourier transforms of the cross-correlation and the autocorrelation signals, respectively. For instance, if the signals are functions of time, the cross-correlation is a measure of the similarity of the two signals as a function of the time lag relative to each other and the autocorrelation is a measure of the similarity of each signal with itself in different instants of time. In this case the coherence is a function of frequency. Analogously, if
x
(
t
)
{\displaystyle x(t)}
and
y
(
t
)
{\displaystyle y(t)}
are functions of space, the cross-correlation measures the similarity of two signals in different points in space and the autocorrelations the similarity of the signal relative to itself for a certain separation distance. In that case, coherence is a function of wavenumber (spatial frequency). The coherence varies in the interval
0
≤
γ
x
y
2
(
f
)
≤
1
{\displaystyle 0\leq \gamma _{xy}^{2}(f)\leq 1}
. If
γ
x
y
2
(
f
)
=
1
{\displaystyle \gamma _{xy}^{2}(f)=1}
it means that the signals are perfectly correlated or linearly related and if
γ
x
y
2
(
f
)
=
0
{\displaystyle \gamma _{xy}^{2}(f)=0}
they are totally uncorrelated. If a linear system is being measured,
x
(
t
)
{\displaystyle x(t)}
being the input and
y
(
t
)
{\displaystyle y(t)}
the output, the coherence function will be unitary all over the spectrum. However, if non-linearities are present in the system the coherence will vary in the limit given above.
== Coherence and correlation == The coherence of two waves expresses how well correlated the waves are as quantified by the cross-correlation function. Cross-correlation quantifies the ability to predict the phase of the second wave by knowing the phase of the first. As an example, consider two waves perfectly correlated for all times (by using a monochromatic light source). At any time, the phase difference between the two waves will be constant. If, when they are combined, they exhibit perfect constructive interference, perfect destructive interference, or something in-between but with constant phase difference, then it follows that they are perfectly coherent. As will be discussed below, the second wave need not be a separate entity. It could be the first wave at a different time or position. In this case, the measure of correlation is the autocorrelation function (sometimes called self-coherence). Degree of correlation involves correlation functions.
== Examples of wave-like states == These states are unified by the fact that their behavior is described by a wave equation or some generalization thereof.