The coherence length can be used for quantifying the degree of temporal (not spatial!) coherence as the propagation length (and thus propagation time) over which coherence degrades significantly, i.e., the optical phase undergoes substantial random changes. It is defined as the coherence time times the vacuum velocity of light.
For example, if the coherence length of a laser is 1 km, the phase relationship between two points along the laser beam, with a distance of 1 km between them, is still significant, but already substantially degraded by phase fluctuations of the laser.
The reason for often using the term coherence length instead of coherence time is that the optical time delays involved in some experiment are often determined by optical path lengths. For example, the interferometer in Figure 1 shows pronounced interference fringes only if the coherence length of the laser light is at least as long as the path-length difference of the two arms. Also, in a setup for making holographic recordings, coherence between two beams with a somewhat different optical path length is required, so that the coherence length of the light source should be longer than the maximum occurring path-length difference. In addition to holography, a number of other applications may require a certain coherence length; see the article on coherence.
where <$\Delta \nu$> is the (full width at half-maximum) linewidth (optical bandwidth). This coherence length is the propagation length after which the magnitude of the coherence function has dropped to the value of <$1 / e$>.
In the literature, one often finds the above equation without the factor <$\pi$> in the denominator – often just for estimating the order of magnitude of the coherence length, without referring to a precise definition of optical bandwidth (e.g., as full width at half-maximum) and coherence length.
Note that such relations are not valid in cases where the coherence function has a more complicated shape, as is the case for, e.g., a frequency comb. Generally, the shape and width of the optical spectrum alone does not fully determine the coherence properties.
Lasers with Long Coherence Lengths
Some lasers, particularly single-frequency solid-state lasers subject to certain methods of laser stabilization, can have very long coherence lengths. For example, 9.5 km coherence length results for a Lorentzian spectrum with a linewidth of 10 kHz. The theoretical limit is set by the Schawlow–Townes linewidth, based on unavoidable quantum noise. The quantum noise influence is weak (allowing for a long coherence length) when the circulating power in the laser is high, the resonator losses per round trip are low, and the round-trip time is long. However, additional technical noise sources, e.g. mechanical vibrations, are normally dominating in practice.
For monolithic semiconductor lasers (laser diodes), even when operating in a single-frequency mode, the coherence length is typically far shorter than for diode-pumped solid-state lasers. The coherence length is limited by phase noise which can result from, e.g., spontaneous emission in the gain medium, and that effect is relatively strong for low-power laser diodes with a short laser resonator and strong output coupling. The temporal coherence is further reduced by a coupling between intensity noise and phase noise (see the article on the linewidth enhancement factor).
Some laser systems used for optical clocks are stabilized to an emission linewidth well below one second, related to a coherence length of over 300 000 km. These are quite delicate systems, but they can be operated stably over long times when being properly engineered.
Coherence Length in Nonlinear Optics
An unfortunate use of the term coherence length is common in nonlinear optics: for example, in second-harmonic generation, the coherence length is often understood as the length over which fundamental and harmonic wave get out of phase (more precisely, the phase difference accumulated over this length is <$\pi$>). This is inconsistent with the general notion of coherence because a predictable phase relationship (strong phase correlation) is definitely maintained over more than this length, although there is a systematic evolution of the relative phase.
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