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
- [math]\displaystyle{ L = \frac{ c }{\, n\, \mathrm{\Delta} f \,} \approx \frac{ \lambda^2 }{\, n\, \mathrm{\Delta} \lambda \,} ~, }[/math]
where [math]\displaystyle{ \, c \, }[/math] is the speed of light in vacuum, [math]\displaystyle{ \, n \, }[/math] is the refractive index of the medium, and [math]\displaystyle{ \, \mathrm{\Delta} f \, }[/math] is the bandwidth of the source or [math]\displaystyle{ \, \lambda \, }[/math] is the signal wavelength and [math]\displaystyle{ \, \Delta \lambda \, }[/math] 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 [math]\displaystyle{ \, L \, }[/math] is given by
- [math]\displaystyle{ L = \frac{\, 2 \ln 2 \,}{ \pi } \, \frac{ \lambda^2 }{\, n_g \, \mathrm{\Delta} \lambda \,}~, }[/math][1][2]
where [math]\displaystyle{ \, \lambda \, }[/math] is the central wavelength of the source, [math]\displaystyle{ n_g }[/math] is the group refractive index of the medium, and [math]\displaystyle{ \, \mathrm{\Delta} \lambda \, }[/math] is the (FWHM) spectral width of the source. If the source has a Gaussian spectrum with FWHM spectral width [math]\displaystyle{ \mathrm{\Delta} \lambda }[/math], then a path offset of [math]\displaystyle{ \, \pm L \, }[/math] 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 [math]\displaystyle{ \, \frac{1}{\, e \,} \approx 37\% \, }[/math] fringe visibility,[3] where the fringe visibility is defined as
- [math]\displaystyle{ V = \frac{\; I_\max - I_\min \;}{ I_\max + I_\min} ~, }[/math]
where [math]\displaystyle{ \, I \, }[/math] 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
References
- ↑ 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. Bibcode: 2002ApOpt..41.5256A. "equation 8".
- ↑ Izatt; Choma; Dhalla (2014). "Theory of Optical Coherence Tomography". Optical Coherence Tomography. Springer Berlin Heidelberg. ISBN 978-3-319-06419-2.
- ↑ Ackermann, Gerhard K. (2007). Holography: A Practical Approach. Wiley-VCH. ISBN 978-3-527-40663-0.
- ↑ "Sam's Laser FAQ - Diode Lasers". https://www.repairfaq.org/sam/laserdio.htm#diobcc4.
- ↑ Tolansky, Samuel (1973). An Introduction to Interferometry. Longman. ISBN 9780582443334.
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