Distance over which a propagating wave maintains a certain degree of coherence
In
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
where is the speed of light in vacuum, is the
medium
, and
is the
bandwidth of the source or
is the signal wavelength and
is the width of the range of wavelengths in the signal.
In optical
Gaussian
emission spectrum, the roundtrip coherence length
is given by
- [1][2]
where is the central wavelength of the source, is the group
medium
, and
is the (FWHM)
spectral width of the source. If the source has a Gaussian spectrum with
FWHM spectral width
, then a path offset of
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
laser beam
which corresponds to
fringe visibility,
[3] where the fringe visibility is defined as
where is the fringe intensity.
In long-distance
.
Lasers
Multimode
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
References