Editing Coherence (physics) (section)

==Temporal coherence==<!--Temporal coherence redirects here-->
[[File:Single frequency correlation.svg|thumb|450px|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 <math>\tau</math> (blue). The coherence time of the wave is infinite since it is perfectly correlated with itself for all delays <math>\tau</math>.<ref name="GerryKnight2005">{{cite book|author1=Christopher Gerry |author2=Peter Knight|title=Introductory Quantum Optics|year=2005|publisher=Cambridge University Press|isbn=978-0-521-52735-4}}</ref>{{rp|p=118}}]]
[[File:phase drift.png|thumb|450px|right|Figure 2: The amplitude of a wave whose phase drifts significantly in time <math>\tau_\mathrm{c}</math> as a function of time ''t'' (red) and a copy of the same wave delayed by <math>2\tau</math>(green).<ref>This figure needs to be changed because, in this figure, the green wave is actually ''not'' a copy of the red wave; both are monochromatic waves with slightly different frequencies. A proper figure would be a combination of a chirp wave and its delayed copy to match the figure and the current figure description.</ref> 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.]]

Temporal coherence is the measure of the average correlation between the value of a wave and itself delayed by <math>\tau</math>, 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]] <math>\tau_\mathrm{c}</math>. At a delay of <math>\tau=0</math> the degree of coherence is perfect, whereas it drops significantly as the delay passes <math>\tau=\tau_\mathrm{c}</math>. The [[coherence length]] <math>L_\mathrm{c}</math> is defined as the distance the wave travels in time <math>\tau_\mathrm{c}</math>.<ref name="Hecht2002">{{cite book |last =Hecht |first=Eugene | title=Optics |year=2002| location=United States of America | publisher=Addison Wesley| edition= 4th| isbn=978-0-8053-8566-3 | language=en}}</ref>{{rp|pp=560, 571–573}}

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 <math>\tau_\mathrm{c}</math> is):{{R|Hecht2002|pages=358-359, 560}}

:<math>\tau_c \Delta f \gtrsim 1.</math>

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.{{R|Hecht2002|page=572}}

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&nbsp;m.<ref name=saleh-teich>{{cite book| last=Saleh | first = Teich| title=Fundamentals of Photonics |publisher=Wiley}}</ref> Not all lasers have a high monochromaticity, however (e.g. for a mode-locked [[Ti-sapphire laser]], Δλ ≈ 2&nbsp;nm – 70&nbsp;nm).

LEDs are characterized by Δλ ≈ 50&nbsp;nm, and tungsten filament lights exhibit Δλ ≈ 600&nbsp;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 packet]]s) 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 ===
[[File:wave packets.png|thumb|400px|right|Figure 3: The amplitude of a wavepacket whose amplitude changes significantly in time <math>\tau_\mathrm{c}</math> (red) and a copy of the same wave delayed by <math>2\tau</math>(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.]]
[[File:interference finite coherence.png|thumb|390px|right|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.]]

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 <math>\tau</math>. A detector measures the time-averaged [[intensity (physics)|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 <math>\tau</math>. 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 <math>2\tau</math>, an infinitely fast detector would measure an intensity that fluctuates significantly over a time ''t'' equal to <math>\tau</math>. In this case, to find the temporal coherence at <math>2\tau_\mathrm{c}</math>, one would manually time-average the intensity.
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