Coherence length

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 ${\displaystyle \mathrm{\Delta }\lambda }$ 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

${\displaystyle L=\frac{2\ln 2}{\pi }\frac{{\lambda }^2}{n_g\mathrm{\Delta }\lambda },}$

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%. 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 ${\displaystyle \frac{1}{e}\approx 37\mathrm{\%}}$ fringe visibility, 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.