# Coherence length

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

${\displaystyle L={\frac {c}{\,n\,\mathrm {\Delta } f\,}}\approx {\frac {\lambda ^{2}}{\,n\,\mathrm {\Delta } \lambda \,}}~,}$

where ${\displaystyle \,c\,}$ is the speed of light in vacuum, ${\displaystyle \,n\,}$ is the refractive index of the medium, and ${\displaystyle \,\mathrm {\Delta } f\,}$ 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 and optical coherence tomography (OCT), assuming that the source has a Gaussian emission spectrum, the roundtrip coherence length ${\displaystyle \,L\,}$ is given by

${\displaystyle L={\frac {\,2\ln 2\,}{\pi }}\,{\frac {\lambda ^{2}}{\,n_{g}\,\mathrm {\Delta } \lambda \,}}~,}$[1][2]

where ${\displaystyle \,\lambda \,}$ is the central wavelength of the source, ${\displaystyle n_{g}}$ is the group refractive index of the medium, and ${\displaystyle \,\mathrm {\Delta } \lambda \,}$ is the (FWHM) spectral width of the source. If the source has a Gaussian spectrum with FWHM spectral width ${\displaystyle \mathrm {\Delta } \lambda }$, then a path offset of ${\displaystyle \,\pm L\,}$ 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 ${\displaystyle \,{\frac {1}{\,e\,}}\approx 37\%\,}$ fringe visibility,[3] where the fringe visibility is defined as

${\displaystyle V={\frac {\;I_{\max }-I_{\min }\;}{I_{\max }+I_{\min }}}~,}$

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 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

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.