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Coherence Time

Definition: a measure of temporal coherence, expressed as the time over which the field correlation decays

German: Kohärenzzeit

Category: general optics

Formula symbol: τcoh

Units: s

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The coherence time can be used for quantifying the degree of temporal coherence of light. In coherence theory, it is essentially defined as the time over which the field correlation function decays. For a stationary light field (having constant statistical properties), the complex degree of temporal coherence can be defined as

coherence function

where E(t) is the complex electric field at a certain location. The complex degree of coherence is 1 for τ = 0 and usually decays monotonically for larger time delays τ. If it decays exponentially (as it occurs e.g. for a laser with quantum noise influences only), one may take the exponential time constant as the coherence time; it is then the value where the coherence decays to e−1 (≈37%) of its value at τ = 0. More generally, the coherence time can be defined by

coherence time

for an arbitrary shape of the coherence function, and in the case of an exponential coherence decay this is the same as the exponential decay time.

Instead of the coherence time, it is common to specify the coherence length, which is simply the coherence time times the vacuum velocity of light, and thus also quantifies temporal (rather than spatial) coherence.

temporal coherence
Figure 1: Animated illustration of temporal coherence. An ensemble of 100 oscillators with nominally the same frequency but with same phase noise (corresponding to white frequency noise) are started at t = 0. The phase space diagram (upper part) shows how they get increasingly out of phase. The thick point shows the complex average amplitude; its oscillations become weaker with time. (This phenomenon is analogous to a reduced interference contrast.) The lower diagram shows the decay of the coherence function with time.

Knowledge of the coherence time (i.e., a single number) can be useful when the shape of the coherence function (or the shape of the Fourier spectrum) is approximately known. Obviously, however, the specification of the coherence time (or the linewidth) alone does not constitute a full characterization of the coherence.

The coherence time is intimately linked with the linewidth of the radiation, i.e., the width of its spectrum. In the case of an exponential coherence decay as above, the optical spectrum has a Lorentzian shape, and the (full width at half-maximum) linewidth is

relation of linewidth and coherence time

The constant factor in this equation (here: 1 / π) is in general different for other shapes of the coherence function (e.g. roughly twice as high for a Gaussian shape). Conversely, the linewidth can be used for estimating the coherence time, but the conversion depends on the spectral shape.

Many authors state that the bandwidth is just the inverse of the coherence time without referring to any concrete definitions of coherence time and bandwidth, nor to a specific spectral shape. This then simply means that the bandwidth is of the order of the inverse coherence time for typical spectral shapes.

In cases where the frequency noise spectrum is not flat but rises strongly at small noise frequencies, there can be a significant degree of coherence even for time delays well above the inverse linewidth; this issue is important, e.g., in the context of self-heterodyne linewidth measurement.

Lasers, particularly single-frequency solid-state lasers, can have long coherence times, compared with the duration of an optical cycle; values of several milliseconds are possible. Even much longer coherence times (as required e.g. in optical frequency metrology) are possible for carefully stabilized lasers; see the article on the stabilization of lasers. A long coherence time is important for many applications (see the article on coherence).

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Bibliography

[1]B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, John Wiley & Sons, Inc., New York (1991)

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See also: coherence, coherence length, linewidth, laser speckle
and other articles in the category general optics

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