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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=cnΔf=λ2nΔλ,{displaystyle L={c over n,Delta f}={lambda ^{2} over nDelta lambda },}

where c{displaystyle c} is the speed of light in a vacuum, n{displaystyle n} is the refractive index of the medium, and Δf{displaystyle Delta f} is the bandwidth of the source or λ{displaystyle lambda } is the signal wavelength and Δλ{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 L{displaystyle L} is given by ^[1]

L=2ln⁡2πλ2nΔλ,{displaystyle L={sqrt {2ln 2 over pi }}{lambda ^{2} over nDelta lambda },}

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

V=Imax−IminImax+Imin,{displaystyle V={I_{max }-I_{min } over I_{max }+I_{min }},,}

where I{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

The coherence length of a mercury-vapor lamp is 0.03 cm.^[4]

See also

• Coherence time


• Superconducting coherence length


• Spatial coherence

References

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