# Coherence Time

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

German: Kohärenzzeit

Units: s

Formula symbol: <$\tau_\textrm{coh}$>

Author: Dr. Rüdiger Paschotta

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

$$g(\tau ) = \frac{{\left\langle {E^*(t) \: E(t + \tau )} \right\rangle }}{{\left\langle {E^*(t) \: E(t)} \right\rangle }}$$where *E(t)* is the complex electric field at a certain location. The complex degree of coherence is 1 for <$\tau = 0$> and usually decays monotonically for larger time delays <$\tau$>. 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 <$\tau = 0$>. More generally, the coherence time can be defined by

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. A measurement of coherence time may be based on that equation, where the coherence function is obtained via a path length dependent interference contrast.

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.

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

$$\Delta \nu = \frac{1}{{\pi \;{\tau _{{\rm{coh}}}}}}$$The constant factor in this equation (here: <$1 / \pi$>) 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).

### Bibliography

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

See also: coherence, coherence length, linewidth, laser speckle

## Questions and Comments from Users

2021-11-03

If I calculate the coherence time as the inverse of the frequency bandwidth, it is several times longer than the pulse duration – can that be?

The author's answer:

If you use the formula of this article, you get a coherence time which is <$\pi$> times smaller, and similar to the FWHM pulse duration. Some people just take the inverse of the bandwidth, but that's just a rough estimate.

2022-01-23

Can the coherence time of a pulsed laser be (much) longer than the FWHM pulse duration?

The author's answer:

Yes, in a sense, if subsequent pulses are mutually coherent. However, in such a situation coherence is not well characterized simply by a coherence time.

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2021-11-03

Can a source that is not monochromatic not be time coherent? For instance, if we have two different well defined wavelengths, the source is not time coherent?

The author's answer:

The coherence time as explained in the article will be infinitely large, as the degree of temporal coherence does not vanish over time. In that sense, such light is fully coherent.