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{{merge|coherence time|target=Coherence length and time|date=July 2013}}
In [[physics]], '''coherence length''' is the [[wave propagation|propagation]] distance over which a [[coherence (physics)|coherent]] wave (e.g. an [[electromagnetic wave]]) maintains a specified [[degree of coherence]]. [[Interference (wave propagation)|Wave interference]] is strong when the paths taken by all of the interfering waves differ by less than the coherence length. A wave with a longer coherence length is closer to a perfect sinusoidal wave. Coherence length is important in [[holography]] and [[telecommunications engineering]].


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This article focuses on the coherence of [[Classical physics|classical]] electromagnetic fields. In [[quantum mechanics]], there is a mathematically analogous concept of the [[Coherence (physics)#Quantum_coherence|quantum coherence]] length of a [[wave function]].
 
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In radio-band systems, the coherence length is approximated by
 
 
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:<math>L={c \over n\, \Delta f},</math>
 
 
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where ''c'' is the speed of light in a vacuum, ''n'' is the [[refractive index]] of the [[Medium (optics)|medium]], and <math>\Delta f</math> is the [[Bandwidth (signal processing)|bandwidth]] of the source.
 
 
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In optical [[information transfer|communications]], the coherence length <math>L</math> is given by
 
 
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:<math>L={2 \ln(2) \over \pi n} {\lambda^2 \over \Delta\lambda},</math>
 
 
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where <math>\lambda</math> is the central [[wavelength]] of the source, <math>n</math> is the [[refractive index]] of the [[Medium (optics)|medium]], and <math>\Delta\lambda</math> is the [[spectral width]] of the source. If the source has a Gaussian spectrum with [[Full width at half maximum|FWHM]] spectral width <math>\Delta\lambda</math>, then a path offset of ±<math>L</math> will reduce the [[fringe visibility]] to 50%.
 
 
</ul>
''Coherence length'' is usually applied to the optical regime.
 
The expression above is a frequently used approximation. Due to ambiguities in the definition of spectral width of a source, however, the following definition of coherence length has been suggested:
 
The coherence length can be measured using a [[Michelson interferometer]] and is the [[optical path length]] difference of a self-interfering [[laser beam]] which corresponds to a <math>1/e=37%</math> fringe visibility,<ref>{{cite book |last=Ackermann |first=Gerhard K. |year=2007 |title=Holography: A Practical Approach |publisher=Wiley-VCH |isbn=3-527-40663-8}}</ref> where the fringe visibility is defined as
 
:<math>V = {I_\max - I_\min \over I_\max + I_\min} ,\,</math>
 
where <math>I</math> is the fringe intensity.
 
In long-distance [[transmission (telecommunications)|transmission]] systems, the coherence length may be reduced by propagation factors such as [[dispersion (optics)|dispersion]], [[scattering]], and [[diffraction]].
 
==Lasers==
 
Multimode [[helium-neon laser]]s have a typical coherence length of 20&nbsp;cm, while the coherence length of singlemode ones can exceed 100&nbsp;m. [[Semiconductor laser]]s reach some 100&nbsp;m. Singlemode [[fiber laser]]s with linewidths of a few kHz can have coherence lengths exceeding 100&nbsp;km. Similar coherence lengths can be reached with optical [[frequency comb]]s due to the narrow linewidth of each tooth. Non-zero visibility is present only for short intervals of pulses repeated after cavity length distances up to this long coherence length.
 
==See also==
{{Portal|Physics}}
* [[Coherence time]]
* [[Superconducting coherence length]]
* [[Spatial coherence]]
 
==References==
{{Reflist|2}}
*{{FS1037C MS188}}
 
{{DEFAULTSORT:Coherence Length}}
[[Category:Electromagnetic radiation]]
[[Category:Optics]]
[[Category:Waves]]

Revision as of 20:22, 3 February 2014



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This article focuses on the coherence of classical electromagnetic fields. In quantum mechanics, there is a mathematically analogous concept of the quantum coherence length of a wave function.

Formulas

In radio-band systems, the coherence length is approximated by

L=cnΔf,

where c is the speed of light in a vacuum, n is the refractive index of the medium, and Δf is the bandwidth of the source.

In optical communications, the coherence length L is given by

L=2ln(2)πnλ2Δλ,

where λ is the central wavelength of the source, n is the refractive index of the medium, and Δλ is the spectral width of the source. If the source has a Gaussian spectrum with FWHM spectral width Δλ, then a path offset of ±L will reduce the fringe visibility to 50%.

Coherence length is usually applied to the optical regime.

The expression above is a frequently used approximation. Due to ambiguities in the definition of spectral width of a source, however, the following definition of coherence length has been suggested:

The coherence length can be measured using a Michelson interferometer and is the optical path length difference of a self-interfering laser beam which corresponds to a 1/e=37% fringe visibility,[1] where the fringe visibility is defined as

V=ImaxIminImax+Imin,

where I is the fringe intensity.

In long-distance transmission systems, the coherence length may be reduced by propagation factors such as dispersion, scattering, and diffraction.

Lasers

Multimode helium-neon lasers have a typical coherence length of 20 cm, while the coherence length of singlemode ones can exceed 100 m. Semiconductor lasers reach some 100 m. Singlemode fiber lasers with linewidths of a few kHz can have coherence lengths exceeding 100 km. Similar coherence lengths can be reached with optical frequency combs due to the narrow linewidth of each tooth. Non-zero visibility is present only for short intervals of pulses repeated after cavity length distances up to this long coherence length.

See also

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