Coherence length

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In physics, coherence length is the propagation distance over which a coherent wave (e.g. an electromagnetic wave) maintains a specified degree of coherence. 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 electromagnetic fields. In quantum mechanics, there is a mathematically analogous concept of the quantum coherence length of a wave function.


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

where is the speed of light in vacuum, is the refractive index of the medium, and is the bandwidth of the source or is the signal wavelength and is the width of the range of wavelengths in the signal.

In optical communications and optical coherence tomography (OCT), assuming that the source has a Gaussian emission spectrum, the roundtrip coherence length is given by


where is the central wavelength of the source, is the group refractive index of the medium, and is the (FWHM) spectral width of the source. If the source has a Gaussian spectrum with FWHM spectral width , then a path offset of 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 fringe visibility,[3] where the fringe visibility is defined as

where is the fringe intensity.

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


Multimode helium–neon lasers have a typical coherence length on the order of centimeters, while the coherence length of longitudinally single-mode lasers can exceed 1 km. Semiconductor lasers can reach some 100 m, but small, inexpensive semiconductor lasers have shorter lengths, with one source[4] claiming 20 cm. 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.

Other light sources[edit]

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 mm for each line by itself.[5] 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[edit]


  1. ^ Akcay, C.; Parrein, P.; Rolland, J.P. (2002). "Estimation of longitudinal resolution in optical coherence imaging". Applied Optics. 41 (25): 5256–5262. Bibcode:2002ApOpt..41.5256A. doi:10.1364/ao.41.005256. PMID 12211551. equation 8
  2. ^ Izatt; Choma; Dhalla (2014). "Theory of Optical Coherence Tomography". In Drexler; Fujimoto (eds.). Optical Coherence Tomography. Springer Berlin Heidelberg. ISBN 978-3-319-06419-2.
  3. ^ Ackermann, Gerhard K. (2007). Holography: A Practical Approach. Wiley-VCH. ISBN 978-3-527-40663-0.
  4. ^ "Sam's Laser FAQ - Diode Lasers". Retrieved 2017-02-06.
  5. ^ Tolansky, Samuel (1973). An Introduction to Interferometry. Longman. ISBN 9780582443334.