Talìa lu còdici di Cuirenza (fisica)

Nta a [[fisica]], a '''coirenza''' eni na prupritati di l'ondi, ca abbilita l'intirfirenza stazziunaria (pa isempiu na custanti timpurali e spazziali).<br>
Chiu giniralmenti a cuirenza addiscrivi tutti li prupritati di currilazzioni tra li quantitati fisichi di n'onda.

Quannu antirfirisciunu, l'ondi iunciunu o lieunu distruttivamenti, a sicunna di a rilativa fasi ca cianu.<br>
Dui ondi si riciuni cuirenti si cianu a stissa fasi rilativa di custanti, ca imprica puri ca cianu a stissa friquenza.<br>
Lu gradu di cuirenza eni misurata da visibbilitati d'intirfirenza, na misura di comu pirfittam l'ondi si ponu cancillari duvutu a n'intirfirenza distruttiva.
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Di siri tradottu di l'angrisi
==Introduction==
Coherence was originally introduced in connection with 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 physics]]. The property of coherence is the basis for commercial applications such as [[holography]], the [[Sagnac interferometer|Sagnac]] [[gyroscope]], [[radio]] [[Phased array|antenna array]]s, [[optical coherence tomography]] and telescope interferometers ([[Interferometry# Astronomical optical interferometry|astronomical optical interferometers]] and [[radio telescope]]s).

==Coherence and correlation==
The coherence of two waves follows from how well correlated the waves are as quantified by the [[cross-correlation]] function<ref name="winter">
{{cite book
|author=Rolf G. Winter
|coauthors=Aephraim M. Steinberg
|title=Coherence
|publisher=AccessScience@McGraw-Hill
|url=http://www.accessscience.com
|doi=10.1036/1097-8542.146900
|unused_data=|last modified= October 24
year=2001
}}</ref><ref name="BornWolf">{{cite book
|author=M.Born
|coauthors=E. Wolf
|title=Principles of Optics
|edition=7th ed.
|year=1999}}</ref><ref name="Loudon">{{cite book
|author=Loudon, Rodney
|title=The Quantum Theory of Light
|publisher=Oxford University Press
|year=2000
|isbn=0-19-850177-3
}}</ref><ref name="mandel">{{cite book
|author=Leonard Mandel
|couthor=Emil Wolf
|title=Optical Coherence and Quantum Optics
|publisher=Cambridge University Press
|year=1995
|isbn=0521417112
}}</ref><ref name="mar">{{cite book
|author=Arvind Marathay
|title=Elements of Optical Coherence Theory
|publisher=John Wiley & Sons Inc
|year=1982
|isbn=0471567892
}}</ref>. The cross-correlation quantifies the ability to predict the value of the second wave by knowing the value of the first. As an example, consider two waves perfectly correlated for all times. At any time, if the first wave changes, the second will change in the same way. If combined they can exhibit complete constructive interference at all times. 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, sometimes called '''self-coherence''', the measure of correlation is the [[autocorrelation]] function.

==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.
*Waves in a rope (up and down) or [[slinky]] (compression and expansion)
*[[Surface waves]] in a liquid
*Electric signals (fields) in transmission cables
*[[Sound]]
*[[Radio]] and [[Microwaves]]
*[[Light]] ([[optics]])
*Electrons, atoms, and any other object (as described by [[quantum physics]]) 

In most of these systems, 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, we measure the [[intensity (physics)|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==
[[Image:single frequency.png|thumb|450px|right|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 τ(green). The coherence time of the wave is infinite since it is perfectly correlated with itself for all delays τ.]]
[[Image:phase drift.png|thumb|450px|right|Figure 2: The amplitude of a wave whose phase drifts significantly in time τ<sub>c</sub> as a function of time ''t'' (red) and a copy of the same wave delayed by 2τ<sub>c</sub>(green). 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 at any pair of times, separated by delay τ. 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<sub>c</sub>''. At τ=0 the degree of coherence is perfect whereas it drops significantly by delay ''τ<sub>c</sub>''. The [[coherence length]] ''L<sub>c</sub>'' is defined as the distance the wave travels in time τ<sub>c</sub>.   

One should be careful not to confuse the coherence time with the time duration of the signal, nor the coherence length with the coherence area (see below).

===The relationship between coherence time and bandwidth===
Since period is the inverse of frequency, it follows that the faster a wave decorrelates (and hence the smaller τ<sub>c</sub> is) the larger the range of frequencies Δf the wave contains. Thus there is a tradeoff:

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

In terms of wavelength (<math>f \lambda=c</math>) this relationship becomes,

:<math>\frac{L_c \Delta \lambda}{\lambda^2} \approx 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]].

===Examples of temporal coherence===
We consider four examples of temporal coherence. 
*A wave containing only a single frequency (monochromatic) is perfectly correlated at all times according to 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.

The most monochromatic sources are usually [[laser]]s; such high monochromaticity implies long coherence lengths (up to hundreds of meters). For example, a stabilized [[helium-neon laser]] can produce light with coherence lengths in excess of 5 m. Not all lasers are monochromatic, however (e.g. for a [[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.

[[Holography]] requires light with a long coherence time. In contrast, [[Optical coherence tomography]] uses light with a short coherence time.

===Measurement of temporal coherence===
[[Image:wave packets.png|thumb|400px|right|Figure 3: The amplitude of a wavepacket whose amplitude changes significantly in time τ<sub>c</sub> (red) and a copy of the same wave delayed by 2τ<sub>c</sub>(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.]]
[[Image: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 τ. A detector measures the time-averaged [[intensity]] of the light exiting the interferometer. The resulting interference visibility (e.g. see Figure 4) gives the temporal coherence at delay τ. 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 2τ<sub>c</sub>, an infinitely fast detector would measure an intensity that fluctuates significantly over a time ''t'' equal to τ<sub>c</sub>. In this case, to find the temporal coherence at 2τ<sub>c</sub>, one would manually time-average the intensity.
<br style="clear:both" />

==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 points in space, ''x<sub>1</sub>'' and ''x<sub>2</sub>'', 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 is called the coherence area, ''A<sub>c</sub>''. This 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. Sometimes people also use “spatial coherence” to refer to the visibility when a wave-like state is combined with a spatially shifted copy of itself.

===Examples of spatial coherence===
<gallery caption="Spatial coherence" perrow=5>
Image:spatial_coherence_infinite_ex1.png|<small>Figure 5: A plane wave with an infinite [[coherence length]].</small>
Image:spatial_coherence_infinite_ex2.png|<small>Figure 6: A wave with a varying profile (wavefront) and infinite coherence length.</small>
Image:spatial_coherence_finite.png|<small>Figure 7: A wave with a varying profile (wavefront) and finite coherence length.</small>
Image:spatial_coherence_pinhole.png|<small>Figure 8: A wave with finite coherence area is incident on a pinhole (small aperture). The wave will [[diffraction|diffract]] out of the pinhole. Far from the pinhole the emerging spherical wavefronts are approximately flat. The coherence area is now infinite while the coherence length is unchanged.</small>
Image:spatial_coherence_detector.png|<small>Figure 9: A wave with infinite coherence area is combined with a spatially-shifted copy of itself. Some sections in the wave interfere constructively and some will interfere destructively. Averaging over these sections, a detector with length D will measure reduced [[interference visibility]]. For example a misaligned [[Mach-Zehnder]] interferometer will do this.</small>
</gallery>

Consider a tungsten light-bulb filament. Different points in the filament emit light independently and have no fixed phase-relationship. In detail, at any point in time the profile of the emitted light is going to be distorted. The profile will change randomly over the coherence time <math>\tau_c</math>. Since for a white-light source such as a light-bulb <math>\tau_c</math> is small, the filament is considered a spatially incoherent source. In contrast, a radio [[Phased array|antenna array]], has large spatial coherence because antennas at opposite ends of the array emit with a fixed phase-relationship. Light waves produced by a laser often have high temporal and spatial coherence (though the degree of coherence depends strongly on the exact properties of the laser). Spatial coherence of laser beams also manifests itself as speckle patterns and diffraction fringes seen at the edges of shadow.

Holography requires temporally and spatially coherent light. Its inventor, Dennis Gabor, produced successful holograms more than ten years before lasers were invented. To produce coherent light he passed the monochromatic light from an emission line of a mercury-vapor lamp through a pinhole spatial filter.

==Spectral coherence==
[[Image:spectral coherence pulse.png|thumb|right|350px|Figure 10: Waves of different frequencies (i.e. colors) interfere to form a pulse if they are coherent.]]
[[Image:spectral coherence continuous.png|thumb|right|350px|Figure 11: Spectrally incoherent light interferes to form continuous light with a randomly varying phase and amplitude]]

Waves of different frequencies (in light these are different colours) can interfere to form a pulse if they have a fixed relative phase-relationship (see [[Fourier transform]]). Conversely, if waves of different frequencies are not coherent, then, when combined, they create a wave that is continuous in time (e.g. white light or [[white noise]]). The temporal duration of the pulse <math>\Delta t</math> is limited by the spectral bandwidth of the light <math>\Delta f</math> according to:

:<math>\Delta f\Delta t \ge 1</math>,

which follows from the properties of the Fourier transform (for quantum particles it also results in the [[Heisenberg uncertainty principle]]).

If the phase depends linearly on the frequency (i.e. <math>\theta (f) \propto f</math>) then the pulse will have the minimum time duration for its bandwidth (a ''transform-limited'' pulse), otherwise it is chirped (see [[Dispersion (optics)|dispersion]]).
===Measurement of spectral coherence===
Measurement of the spectral coherence of light requires a [[nonlinear optics|nonlinear]] optical interferometer, such as an intensity [[optical autocorrelation|optical correlator]], [[frequency-resolved optical gating]] (FROG), or [[Spectral phase interferometry for direct electric-field reconstruction]] (SPIDER).
<br style="clear:both" />

==Polarization coherence==
Light also has a [[polarization]], which is the direction in which the electric field oscillates. Unpolarized light is composed of two equally intense incoherent light waves with orthogonal polarizations. The electric field of the unpolarized light wanders in every direction and changes in phase over the coherence time of the two light waves. A [[polarizer]] rotated to any angle will always transmit half the incident intensity when averaged over time.

If the electric field wanders by a smaller amount the light will be partially polarized so that at some angle, the polarizer will transmit more than half the intensity. If a wave is combined with an orthogonally polarized copy of itself delayed by less than the coherence time, partially polarized light is created. 

The polarization of a light beam is represented by a vector in the [[Polarization#Parameterizing polarization|Poincare sphere]]. For polarized light the end of the vector lies on the surface of the sphere, whereas the vector has zero length for unpolarized light. The vector for partially polarized light lies within the sphere.

==Applications==
===Holography===
Coherent superpositions of ''optical wave fields'' include [[holography]]. Holographic objects are used frequently in daily life in bank notes and credit cards.

===Non-optical wave fields===
Further applications concern the coherent superposition of  ''non-optical wave fields''. In quantum mechanics  for example one considers a probability field, which is related to the wave function <math>\psi (\mathbf r)</math> (interpretation: density of the probability amplitude). Here the applications concern, among others, the future technologies of [[quantum computing]] and the already available technology of [[quantum cryptography]]. Additionally the problems of the following subchapter are treated.

==Quantum coherence and the range limitation problem==
****This section is linked from [[Quantum entanglement]]***
In [[quantum mechanics]], all objects have wave-like properties (see [[De Broglie hypothesis|de Broglie wave]]s). For instance, in Young's double-slit experiment electrons can be used in the place of light waves. Each electron can go through either slit and hence has two paths that it can take to a particular final position. In quantum mechanics these two paths interfere. If there is destructive interference, the electron never arrives at that particular position. This ability to interfere is called quantum coherence.

The quantum description of perfectly coherent paths is called a [[pure state]], in which the two paths are combined in a [[Quantum superposition|superposition]]. The correlation between the two particles exceeds what would be predicted for classical correlation alone (see [[Bell's inequalities]]).  If this two-particle system is decohered (which would occur in a measurement via [[Einselection]]), then there is no longer any phase relationship between the two states.  The quantum description of imperfectly coherent paths is called a [[mixed state]], described by a [[density matrix]] (also called "statistical operator") and is entirely analogous to a classical system of mixed probabilities (the correlations are classical).

Large-scale ([[macroscopic]]) quantum coherence leads to very amazing phenomena. For instance, the [[laser]], [[superconductivity]], and [[superfluidity]] are examples of highly coherent quantum systems.  One example that shows the amazing possibilities of macroscopic quantum coherence is the [[Schrödinger's cat]] thought experiment.   Another example of quantum coherence is in a [[Bose-Einstein condensate]]. Here, all the atoms that make up the condensate are [[in phase|in-phase]]; they are thus necessarily all described by a single quantum wavefunction; their behavior is [[(need a link to page which explains what does 'communal' mean here)|communal]] and [[(need a link to page which explains what does 'inseparable' mean here)|inseparable]] until the coherence is destroyed.

Especially, the coherence is destroyed, if at the end of a long "coherent wave guide", presently realized through  glass fibres, a signal magnification is necessary. The magnification processes usually involve a certain randomness, which hampers coherence and implies a range limitation of methods based on it. This implies, for example, that the methods of quantum cryptography are presently, in 2008, limited to a range of typically 100 km.

==See also==
*[[Atomic coherence]]
*[[Coherence length]]
*[[Coherent state]]
*[[Optical heterodyne detection]]
*[[Quantum decoherence]]
* [[Quantum Zeno effect]]
* [[Measurement problem]]
* [[Measurement in quantum mechanics]]

==References==
<references/>
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