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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Coherence (physics) | 2/4 | https://en.wikipedia.org/wiki/Coherence_(physics) | reference | science, encyclopedia | 2026-05-05T13:41:37.176399+00:00 | kb-cron |
Waves in a rope (up and down) or slinky (compression and expansion) Surface waves in a liquid Electromagnetic signals (fields) in transmission lines Sound Radio waves and microwaves Light waves (optics) Matter waves associated with, for examples, electrons and atoms In system with macroscopic waves, one can measure the wave directly. Consequently, its correlation with another wave can simply be calculated. However, in optics one cannot measure the electric field directly as it oscillates much faster than any detector's time resolution. Instead, one measures the intensity of the light. Most of the concepts involving coherence which will be introduced below were developed in the field of optics and then used in other fields. Therefore, many of the standard measurements of coherence are indirect measurements, even in fields where the wave can be measured directly.
== Temporal coherence ==
Temporal coherence is the measure of the average correlation between the value of a wave and itself delayed by
τ
{\displaystyle \tau }
, at any pair of times. Temporal coherence tells us how monochromatic a source is. In other words, it characterizes how well a wave can interfere with itself at a different time. The delay over which the phase or amplitude wanders by a significant amount (and hence the correlation decreases by significant amount) is defined as the coherence time
τ
c
{\displaystyle \tau _{\mathrm {c} }}
. At a delay of
τ
=
0
{\displaystyle \tau =0}
the degree of coherence is perfect, whereas it drops significantly as the delay passes
τ
=
τ
c
{\displaystyle \tau =\tau _{\mathrm {c} }}
. The coherence length
L
c
{\displaystyle L_{\mathrm {c} }}
is defined as the distance the wave travels in time
τ
c
{\displaystyle \tau _{\mathrm {c} }}
. The coherence time is not the time duration of the signal; the coherence length differs from the coherence area (see below).
=== The relationship between coherence time and bandwidth === The larger the bandwidth – range of frequencies Δf a wave contains – the faster the wave decorrelates (and hence the smaller
τ
c
{\displaystyle \tau _{\mathrm {c} }}
is):
τ
c
Δ
f
≳
1.
{\displaystyle \tau _{c}\Delta f\gtrsim 1.}
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. Narrow bandwidth lasers have long coherence lengths (up to hundreds of meters). For example, a stabilized and monomode helium–neon laser can easily produce light with coherence lengths of 300 m. Not all lasers have a high monochromaticity, however (e.g. for a mode-locked Ti-sapphire laser, Δλ ≈ 2 nm – 70 nm). LEDs are characterized by Δλ ≈ 50 nm, and tungsten filament lights exhibit Δλ ≈ 600 nm, so these sources have shorter coherence times than the most monochromatic lasers.
=== Examples of temporal coherence === Examples of temporal coherence include:
A wave containing only a single frequency (monochromatic) is perfectly correlated with itself at all time delays, in accordance with the above relation. (See Figure 1) Conversely, a wave whose phase drifts quickly will have a short coherence time. (See Figure 2) Similarly, pulses (wave packets) of waves, which naturally have a broad range of frequencies, also have a short coherence time since the amplitude of the wave changes quickly. (See Figure 3) Finally, white light, which has a very broad range of frequencies, is a wave which varies quickly in both amplitude and phase. Since it consequently has a very short coherence time (just 10 periods or so), it is often called incoherent. Holography requires light with a long coherence time. In contrast, optical coherence tomography, in its classical version, uses light with a short coherence time.
=== Measurement of temporal coherence ===
In optics, temporal coherence is measured in an interferometer such as the Michelson interferometer or Mach–Zehnder interferometer. In these devices, a wave is combined with a copy of itself that is delayed by time
τ
{\displaystyle \tau }
. A detector measures the time-averaged intensity of the light exiting the interferometer. The resulting visibility of the interference pattern (e.g. see Figure 4) gives the temporal coherence at delay
τ
{\displaystyle \tau }
. Since for most natural light sources, the coherence time is much shorter than the time resolution of any detector, the detector itself does the time averaging. Consider the example shown in Figure 3. At a fixed delay, here
2
τ
{\displaystyle 2\tau }
, an infinitely fast detector would measure an intensity that fluctuates significantly over a time t equal to
τ
{\displaystyle \tau }
. In this case, to find the temporal coherence at
2
τ
c
{\displaystyle 2\tau _{\mathrm {c} }}
, one would manually time-average the intensity.
== Spatial coherence == In some systems, such as water waves or optics, wave-like states can extend over one or two dimensions. Spatial coherence describes the ability for two spatial points x1 and x2 in the extent of a wave to interfere when averaged over time. More precisely, the spatial coherence is the cross-correlation between two points in a wave for all times. If a wave has only 1 value of amplitude over an infinite length, it is perfectly spatially coherent. The range of separation between the two points over which there is significant interference defines the diameter of the coherence area,
A
c
{\displaystyle A_{\mathrm {c} }}
(Coherence length
l
c
{\displaystyle l_{\mathrm {c} }}
, often a feature of a source, is usually an industrial term related to the coherence time of the source, not the coherence area in the medium).
A
c
{\displaystyle A_{\mathrm {c} }}
is the relevant type of coherence for the Young's double-slit interferometer. It is also used in optical imaging systems and particularly in various types of astronomy telescopes. A distance
z
{\displaystyle z}
away from an incoherent source with surface area
A
s
{\displaystyle A_{\mathrm {s} }}
,
A
c
=
λ
2
z
2
A
s
{\displaystyle A_{\mathrm {c} }={\frac {\lambda ^{2}z^{2}}{A_{\mathrm {s} }}}}