Editing Coherence length

{{Short description|Distance over which a propagating wave maintains a certain degree of coherence}}
{{Use American English|date = March 2019}}
In [[physics]], '''coherence length''' is the 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]].

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]].

==Formulas==

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

:<math>L = \frac{ c }{\, n\, \mathrm{\Delta} f \,} \approx \frac{ \lambda^2 }{\, n\, \mathrm{\Delta} \lambda \,} ~,</math>

where <math>\, c \,</math> is the speed of light in vacuum, <math>\, n \,</math> is the [[refractive index]] of the [[Medium (optics)|medium]], and <math>\, \mathrm{\Delta} f \,</math> is the [[Bandwidth (signal processing)|bandwidth]] of the source or <math>\, \lambda \,</math> is the signal wavelength and <math>\, \Delta \lambda \,</math> is the width of the range of wavelengths in the signal.

In optical [[information transfer|communications]] and [[optical coherence tomography]] (OCT), assuming that the source has a [[Gaussian]] emission spectrum, the roundtrip coherence length <math>\, L \,</math> is given by 

:<math>L = \frac{\, 2 \ln 2 \,}{ \pi } \, \frac{ \lambda^2 }{\, n_g \, \mathrm{\Delta} \lambda \,}~,</math><ref name=Akcay>{{cite journal |author1=Akcay, C. |author2=Parrein, P. |author3=Rolland, J.P. |year=2002 |title=Estimation of longitudinal resolution in optical coherence imaging |journal=Applied Optics |volume=41 |issue=25 |pages=5256–5262 |quote=equation 8 |doi=10.1364/ao.41.005256|pmid=12211551 |bibcode=2002ApOpt..41.5256A }}</ref><ref name=Drexler>{{cite book |editor-last1=Drexler |editor-last2=Fujimoto |year=2014 |title=Optical Coherence Tomography |chapter=Theory of Optical Coherence Tomography |last1=Izatt |last2=Choma |last3=Dhalla |publisher=Springer Berlin Heidelberg |isbn=978-3-319-06419-2}}</ref>

where <math>\, \lambda \,</math> is the central [[wavelength]] of the source, <math>n_g</math> is the group [[refractive index]] of the [[Medium (optics)|medium]], and <math>\, \mathrm{\Delta} \lambda \,</math> is the (FWHM) [[spectral width]] of the source. If the source has a Gaussian spectrum with [[Full width at half maximum|FWHM]] spectral width <math>\mathrm{\Delta} \lambda</math>, then a path offset of <math>\, \pm L \,</math> will reduce the [[fringe visibility]] to 50%.  It is important to note that this is a roundtrip coherence length — this definition is applied in applications like OCT where the light traverses the measured displacement twice (as in a [[Michelson interferometer]]).  In transmissive applications, such as with a [[Mach–Zehnder interferometer]], the light traverses the displacement only once, and the coherence length is effectively doubled.

The coherence length can also be measured using a Michelson interferometer and is the [[optical path length]] difference of a self-interfering [[laser beam]] which corresponds to <math>\, \frac{1}{\, e \,} \approx 37\% \,</math> fringe visibility,<ref>{{cite book |last=Ackermann |first=Gerhard K. |year=2007 |title=Holography: A Practical Approach |publisher=Wiley-VCH |isbn=978-3-527-40663-0}}</ref> where the fringe visibility is defined as

:<math>V = \frac{\; I_\max - I_\min \;}{ 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 on the order of centimeters, while the coherence length of longitudinally single-mode lasers can exceed 1&nbsp;km. [[Semiconductor laser]]s can reach some 100&nbsp;m, but small, inexpensive semiconductor lasers have shorter lengths, with one source<ref>{{Cite web|url=https://www.repairfaq.org/sam/laserdio.htm#diobcc4|title=Sam's Laser FAQ - Diode Lasers|website=www.repairfaq.org|access-date=2017-02-06}}</ref> claiming up to 20&nbsp;cm, although multi-mode diodes will have even shorter coherence lengths. Singlemode [[fiber laser]]s with [[Laser linewidth|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.

==Other light sources==
Tolansky's ''An introduction to Interferometry'' has a chapter on sources which quotes a line width of around 0.052 angstroms for each of the Sodium D lines in an uncooled low-pressure sodium lamp, corresponding to a coherence length of around 67&nbsp;mm for each line by itself.<ref>{{cite book |title=An Introduction to Interferometry |first=Samuel |last=Tolansky |isbn=9780582443334 |year=1973 |publisher=Longman}}</ref> Cooling the low pressure sodium discharge to [[liquid nitrogen]] temperatures increases the individual D line coherence length by a factor of 6.  A very narrow-band interference filter would be required to isolate an individual D line.

==See also==
{{Portal|Physics}}
* [[Coherence time]]
* [[Superconducting coherence length]]
* [[Spatial coherence]]

==References==
{{reflist|30em}}

* {{FS1037C MS188}}

{{DEFAULTSORT:Coherence Length}}
[[Category:Electromagnetic radiation]]
[[Category:Physical optics]]
[[Category:Waves]]
[[Category:Optical quantities]]