Coherence length

 ( Electromagnetic Radiation ) 

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.

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

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

where $\backslash ,\; c\; \backslash ,$ is the speed of light in vacuum, $\backslash ,\; n\; \backslash ,$ is the refractive index of the medium, and $\backslash ,\; \backslash mathrm\{\backslash Delta\}\; f\; \backslash ,$ is the bandwidth of the source or $\backslash ,\; \backslash lambda\; \backslash ,$ is the signal wavelength and $\backslash ,\; \backslash Delta\; \backslash lambda\; \backslash ,$ 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 $\backslash ,\; L\; \backslash ,$ is given by

$L\; =\; \backslash frac\{\backslash ,\; 2\; \backslash ln\; 2\; \backslash ,\}\{\; \backslash pi\; \}\; \backslash ,\; \backslash frac\{\; \backslash lambda^2\; \}\{\backslash ,\; n\_g\; \backslash ,\; \backslash mathrm\{\backslash Delta\}\; \backslash lambda\; \backslash ,\}~,$

(2025). 9783319064192, Springer Berlin Heidelberg. ISBN 9783319064192

where $\backslash ,\; \backslash lambda\; \backslash ,$ is the central wavelength of the source, $n\_g$ is the group refractive index of the medium, and $\backslash ,\; \backslash mathrm\{\backslash Delta\}\; \backslash lambda\; \backslash ,$ is the (FWHM) spectral width of the source. If the source has a Gaussian spectrum with FWHM spectral width $\backslash mathrm\{\backslash Delta\}\; \backslash lambda$, then a path offset of $\backslash ,\; \backslash pm\; L\; \backslash ,$ 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 $\backslash ,\; \backslash frac\{1\}\{\backslash ,\; e\; \backslash ,\}\; \backslash approx\; 37\backslash \%\; \backslash ,$ fringe visibility,

where the fringe visibility is defined as

$V\; =\; \backslash frac\{\backslash ;\; I\_\backslash max\; -\; I\_\backslash min\; \backslash ;\}\{\; I\_\backslash max\; +\; I\_\backslash min\}\; ~,$

where $\backslash ,\; I\; \backslash ,$ is the fringe intensity.

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

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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 claiming up to 20 cm, although multi-mode diodes will have even shorter coherence lengths. Singlemode with Laser linewidth of a few kHz can have coherence lengths exceeding 100 km. Similar coherence lengths can be reached with optical 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.

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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.

Samuel Tolansky (1973). 9780582443334, Longman. ISBN 9780582443334

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.

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See also

• Coherence time
• Superconducting coherence length
• Spatial coherence

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