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

$L={c \over n\, \Delta f},$

where c is the speed of light in a vacuum, n is the refractive index of the medium, and $\Delta f$ is the bandwidth of the source.

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

$L={\sqrt{2 \ln(2) \over \pi n}} {\lambda^2 \over \Delta\lambda},$

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

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

where $I$ 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 singlemode ones can exceed 100 m. Semiconductor lasers reach some 100 m. 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.

See also

• Coherence time
• Superconducting coherence length
• Spatial coherence

References

1. Akcay, C., Parrein. P., and Rolland, J. P. (2002). Estimation of longitudinal resolution in optical coherence imaging. Applied Optics 41 (25), 5256-5262
2. Ackermann, Gerhard K. (2007). Holography: A Practical Approach. Wiley-VCH. ISBN 3-527-40663-8.

•  This article incorporates public domain material from the General Services Administration document "Federal Standard 1037C" (in support of MIL-STD-188).