# Coherence Length

Definition: a measure of temporal coherence, expressed as the propagation distance over which the coherence significantly decays

German: Kohärenzlänge

Categories: general optics, nonlinear optics

Units: m

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

Author: Dr. Rüdiger Paschotta

Cite the article using its DOI: https://doi.org/10.61835/bix

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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.

## Calculations

For light with a Lorentzian optical spectrum resulting from a random walk of the optical phase, the coherence length can be calculated as

$${L_{{\rm{coh}}}} = c\;{\tau _{{\rm{coh}}}} = \frac{c}{{\pi \;\Delta \nu }}$$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.

See also: coherence, coherence time, linewidth, laser speckle, spotlight 2006-09-22

## Questions and Comments from Users

2020-04-03

If the free running, absolute linewidth of a solid state laser is say 200 kHz over short timescales (such as 100 μs) but then the frequency of the laser drifts associated with changes in temperature, and that results in a larger effective linewidth of several megahertz over longer timescales, how does this effect the coherence length?

The author's answer:

If such drifts are not very fast, they will not affect the coherence length. That will remain limited by the short-term phase fluctuations.

2020-10-15

For a pulsed laser, could the coherence time be shorter than the pulse duration? I have a gain-switched laser diode with a pulse width of the order of 10 ps, but a spectrometer gives me a spectral width of 3 nm, corresponding to fractions of ps coherence time. How can this be explained?

The author's answer:

During the pulse duration, the carrier density in the laser diode changes quite significantly, and that affects the instantaneous frequency of its optical output. Therefore, the generated pulses are strongly chirped, and the coherence time is reduced accordingly.

2020-11-25

How can one measure the coherence length of a laser using a spectrometer?

The author's answer:

A spectrometer can be used to measure the optical linewidth. If this is limited by the coherence length, you may estimate the latter with the first equation mentioned in this article – but you cannot check that way whether that assumption is valid.

2021-12-02

What if we have a multimode laser? Do we have a coherence length for each individual mode or we can calculate the coherence length using the whole spectrum distribution?

The author's answer:

You can determine a coherence length for each individual mode.

Alternatively, you may consider the coherence of the whole laser output, where however the coherence properties are more complicated. Simply calculating the coherence length from the overall bandwidth will not provide an accurate result.

2022-02-14

I understand that coherence length can be theoretically very long, but what are the highest coherence lengths of CW and pulsed lasers that have been successfully manufactured and are actually used in practice?

2022-07-05

Regarding the use of “coherence length” in nonlinear optics: Indeed, in many textbooks and articles the term “coherence length” is used and it is ultra confusing to people new to the field. However, if you check for example Boyd's standard textbook on nonlinear optics, Boyd tries to coin the term “coherent interaction length”, which is a little bit better in terms of avoiding confusion, however this convention/term is often colloquially shortened in many publications to “coherence length”.

The author's answer:

I agree. Thank you very much for the comment.

2023-05-30

Can the coherence length of a laser be shorter than its wavelength?

The author's answer:

No, that's hardly possible. But in extreme cases, it is at least not far longer.

2023-09-29

Is there a relationship that quantify the contrast ratio of an interferometer, based on the optical path delay of the interferometer and coherence length of the laser used?

For instance, what can be the expected contrast if coherence length is equal to OPD? 10 times smaller?

The author's answer:

There are such such equations, but they depend on the spectral shape of the noise spectrum. Often, you get a monotonous decrease of fringe visibility with increasing OPD, and a substantial reduction (but less than a factor 10) if OPD = coherence length.

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2020-03-16

How can one calculate or quantify coherence length from laser phase noise values?

The author's answer:

Laser phase noise may be quantified in different forms – for example, with an autocorrelation function or a power spectral density. One requires some quite involved equations for calculating the coherence length from that; this is a technically quite complicated field. Sorry, I don't have a good reference available at the moment – maybe someone else can help?