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

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

where ${\displaystyle \,c\,}$ is the speed of light in a 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 ${\displaystyle \,\Delta \lambda \,}$ 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 ${\displaystyle \,L\,}$ is given by

${\displaystyle L=C\,{\frac {\lambda ^{2}}{\,n\,\mathrm {\Delta } \lambda \,}}~,}$[1][2]

where ${\displaystyle \,\lambda \,}$ is the central wavelength of the source, ${\displaystyle n}$ is the 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%.

The constant ${\displaystyle \,C\,}$ is roughly 1/ 2 . Some authors give it as ${\textstyle \;{\frac {\,2\ln 2\,}{\pi }}\approx 0.4413}$,[1] while others give it as ${\textstyle \;{\sqrt {{\frac {\,2\ln 2\,}{\pi }}\;}}\approx 0.6643}$.[2]

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 ${\displaystyle \,{\frac {1}{\,e\,}}\approx 37\%\,}$ fringe visibility,[3] 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.

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[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.

References

1. ^ a b Akcay, C.; Parrein, P.; Rolland, J.P. (2002). "Estimation of longitudinal resolution in optical coherence imaging". Applied Optics. 41 (25): 5256–5262. doi:10.1364/ao.41.005256. PMID 12211551. equation 8
2. ^ a b Drexler, Fujimoto (2008). Optical Coherence Tomography. Springer Berlin Heidelberg. ISBN 978-3-540-77549-2.
3. ^ Ackermann, Gerhard K. (2007). Holography: A Practical Approach. Wiley-VCH. ISBN 978-3-527-40663-0.
4. ^ "Sam's Laser FAQ - Diode Lasers". www.repairfaq.org. Retrieved 2017-02-06.
5. ^ Tolansky, Samuel (1973). An Introduction to Interferometry. Longman. ISBN 9780582443334.