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Coherence length

## Summary

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.

## Formulas

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

${\displaystyle L={c \over n\,\Delta f}={\lambda ^{2} \over n\Delta \lambda },}$

where ${\displaystyle c}$ is the speed of light in a vacuum, ${\displaystyle n}$ is the refractive index of the medium, and ${\displaystyle \Delta f}$ is the bandwidth of the source or ${\displaystyle \lambda }$ is the signal wavelength and ${\displaystyle \Delta \lambda }$ is the width of the range of wavelengths in the signal.

In optical communications, assuming that the source has a Gaussian emission spectrum, the coherence length ${\displaystyle L}$ is given by [1]

${\displaystyle L={1 \over 2}L_{FWHM}={2\ln 2 \over \pi }{\lambda ^{2} \over n\Delta \lambda },}$

where ${\displaystyle \lambda }$ is the central wavelength of the source, ${\displaystyle n}$ is the refractive index of the medium, and ${\displaystyle \Delta \lambda }$ is the (FWHM) spectral width of the source. If the source has a Gaussian spectrum with FWHM spectral width ${\displaystyle \Delta \lambda }$, then a path offset of ±${\displaystyle L}$ 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 ${\displaystyle 1/e=37\%}$ fringe visibility,[2] where the fringe visibility is defined as

${\displaystyle V={I_{\max }-I_{\min } \over I_{\max }+I_{\min }},\,}$

where ${\displaystyle I}$ 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 single-mode lasers can exceed 100 m. Semiconductor lasers reach some 100 m, but small, inexpensive semiconductor lasers have shorter lengths, with one source[3] 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

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. 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.