Coherence length explained

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)/(n \Delta f) ≈ (λ ^{2})/(n \Delta λ) ~,

where

c is the speed of light in vacuum, n is the refractive index of the medium, and \Delta f is the bandwidth of the source or λ is the signal wavelength and \Delta λ 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

L is given by L = (2ln2)/(\pi) (λ ^{2})/(n _{g} \Delta λ) ~, ^{[1] [2]}

where

λ is the central wavelength of the source, n _{g} is the group refractive index of the medium, and \Delta λ is the (FWHM) spectral width of the source. If the source has a Gaussian spectrum with FWHM spectral width \Delta λ , then a path offset of \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

(1)/(e) ≈ 37 \% fringe visibility,^[3] where the fringe visibility is defined asV = (I _{max} - I _{min})/(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 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.

See also

• Coherence time
• Superconducting coherence length
• Spatial coherence

Notes and References

1. Akcay, C. . Parrein, P. . Rolland, J.P. . 2002 . Estimation of longitudinal resolution in optical coherence imaging . Applied Optics . 41 . 25 . 5256–5262 . equation 8 . 10.1364/ao.41.005256. 12211551 . 2002ApOpt..41.5256A .
2. Book: Drexler . Fujimoto . 2014 . Optical Coherence Tomography . Theory of Optical Coherence Tomography . Izatt . Choma . Dhalla . Springer Berlin Heidelberg . 978-3-319-06419-2.
3. Book: Ackermann, Gerhard K. . 2007 . Holography: A Practical Approach . Wiley-VCH . 978-3-527-40663-0.
4. Web site: Sam's Laser FAQ - Diode Lasers. www.repairfaq.org. 2017-02-06.
5. Book: Tolansky, Samuel . An Introduction to Interferometry . 9780582443334 . 1973 . Longman.