Coherence length

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 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 and optical coherence tomography (OCT), assuming that the source has a Gaussian emission spectrum, the roundtrip coherence length ${\displaystyle \,L\,}$  is given by

${\displaystyle L={\frac {\,2\ln 2\,}{\pi }}\,{\frac {\lambda ^{2}}{\,n_{g}\,\mathrm {\Delta } \lambda \,}}~,}$ ^[1][2]

where ${\displaystyle \,\lambda \,}$  is the central wavelength of the source, ${\displaystyle n_{g}}$  is the group 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%. 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 ${\displaystyle \,{\frac {1}{\,e\,}}\approx 37\%\,}$  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.

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.

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.

• Coherence time
• Superconducting coherence length
• Spatial coherence

•   This article incorporates public domain material from Federal Standard 1037C. General Services Administration. Archived from the original on 2022-01-22. (in support of MIL-STD-188).