Coherence length

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

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

L = 2 ln ⁡ 2 π λ 2 n g Δ λ   , {\displaystyle L={\frac {\,2\ln 2\,}{\pi }}\,{\frac {\lambda ^{2}}{\,n_{g}\,\mathrm {\Delta } \lambda \,}}~,} {\displaystyle L={\frac {\,2\ln 2\,}{\pi }}\,{\frac {\lambda ^{2}}{\,n_{g}\,\mathrm {\Delta } \lambda \,}}~,}[1][2]

where λ {\displaystyle \,\lambda \,} {\displaystyle \,\lambda \,} is the central wavelength of the source, n g {\displaystyle n_{g}} {\displaystyle n_{g}} is the group refractive index of the medium, and Δ λ {\displaystyle \,\mathrm {\Delta } \lambda \,} {\displaystyle \,\mathrm {\Delta } \lambda \,} is the (FWHM) spectral width of the source. If the source has a Gaussian spectrum with FWHM spectral width Δ λ {\displaystyle \mathrm {\Delta } \lambda } {\displaystyle \mathrm {\Delta } \lambda }, then a path offset of ± L {\displaystyle \,\pm L\,} {\displaystyle \,\pm L\,} will reduce the fringe visibility to 50%. 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 % {\displaystyle \,{\frac {1}{\,e\,}}\approx 37\%\,} {\displaystyle \,{\frac {1}{\,e\,}}\approx 37\%\,} fringe visibility,[3] where the fringe visibility is defined as

V = I max − I min I max + I min   , {\displaystyle V={\frac {\;I_{\max }-I_{\min }\;}{I_{\max }+I_{\min }}}~,} {\displaystyle V={\frac {\;I_{\max }-I_{\min }\;}{I_{\max }+I_{\min }}}~,}

where I {\displaystyle \,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 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 up to 20 cm, although multi-mode diodes will have even shorter coherence lengths. 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

References

  1. Akcay, C.; Parrein, P.; Rolland, J.P. (2002). "Estimation of longitudinal resolution in optical coherence imaging". Applied Optics. 41 (25): 5256–5262. Bibcode:2002ApOpt..41.5256A. doi:10.1364/ao.41.005256. PMID 12211551. equation 8
  2. Izatt; Choma; Dhalla (2014). "Theory of Optical Coherence Tomography". In Drexler; Fujimoto (eds.). Optical Coherence Tomography. Springer Berlin Heidelberg. ISBN 978-3-319-06419-2.
  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.