# Coherence length

In optics, the coherence length is the maximum path length or transit time difference that two light beams from the same source may have, so that when they are superimposed a (spatially and temporally) stable interference pattern is created. ${\ displaystyle l_ {c}}$

In this context, real, non-idealized light sources are considered which do not emit absolutely monochromatic light waves with a temporally constant polarization and phase relationship to one another; with absolutely monochromatic light the coherence length would be infinite. Lasers generate light with a long to very long coherence length (up to many kilometers). In the case of natural light ( sunlight , flame , thermal radiation, etc.), it is in the range of the mean wavelength (order of magnitude 10 −6 m).

## definition

The coherence length corresponds to the optical path length that the light travels during the coherence time: ${\ displaystyle \ tau _ {c}}$

${\ displaystyle l_ {c} = \ tau _ {c} \, {\ frac {c} {n}}}$

It is

## Similar concept in crystallography

Coherence length means the distance up to which the positions of the zero crossings in the wave field can still be reliably predicted if the distance between two neighboring zero crossings is known.

This can be compared with an example from crystallography : if the crystal orientation of a few atoms in the seed crystal and the exact distance between atoms in a single crystal made of silicon, for example , are known, the position of very distant atoms can be predicted exactly, with silicon up to a few meters. This safe distance corresponds to the coherence length.

## example

Above: large coherence length, below: small coherence length

The upper curve shows many regular oscillations between A and B. The path difference in an interference attempt must be shorter than the distance between A and B, so that the beginning and end of this oscillation train overlap and a visible interference pattern is just found.

The oscillation train underneath has a considerably shorter coherence length; it also consists of individual oscillation trains that are separated by phase jumps . Assume that the path length difference of the interference attempt is just as long as the distance DE. Then this wave train does not generate a pattern, especially not shorter ones (such as FG). In contrast, EF and GH can just generate interference patterns. Overall, a poorly visible pattern will result because the interference maxima that appear constantly at random points (for example between the last end of EF and the start of GH with an undefined phase relationship) provide increasing background brightness.

There are several causes for finite coherence lengths:

• In solids there are so many different energy levels in the atomic shell that separate spectral lines can no longer be observed. The coherence length is only in the nanometer range, which, according to Fourier analysis, leads to a very large frequency and wavelength blur .
• Shortly after the start of the “broadcast”, a neighboring atom begins, without prior agreement, its own broadcast on the same frequency with a different phase position. Even if both individual transmissions run undisturbed, there are three phase jumps in total.

Laser light, on the other hand, is considered to be the most easily generated monochromatic light and has the greatest coherence length (up to several kilometers). A helium-neon laser , for example, can produce light with coherence lengths of over 1 km; frequency-stabilized lasers achieve a multiple. However, not all lasers are monochromatic (e.g. titanium-sapphire laser Δλ ≈ 2 nm .. 70 nm). LEDs are less monochromatic (Δλ ≈ 30 nm) and therefore have shorter coherence times than most monochromatic lasers. Since a laser has the same phase over its entire aperture , laser light has a very high spatial coherence.

## Effect of the double gap test

Light rays in the double slit test
Superposition of an oscillation with its copy shifted by Δs

The reason is that the brightness at the right measuring point (target point in the image above x) hardly differs from the brightness of the surroundings. The reason follows from the picture below:

• The upper wave train of relatively short coherence length reaches the measuring point from the direction of the upper gap.
• The lower wave train comes from the same light source and has the same coherence length. However, it arrives at the measuring point a little late because it comes from the lower gap and therefore has to cover a distance that is longer by Δs.
• If the measuring point were to be selected a little higher or lower, Δs would be larger or smaller.

At the measuring point, the elongations (instantaneous deflections) of both wave trains add up; the result can be larger or smaller than the amplitude of each partial wave alone. The red marked periods in the picture mean constructive interference , i.e. maximum brightness. Because of the short coherence length, this is only the case for about 70% of the total time. During the rest of the time, the brightness at the measuring point is lower. Instead, the brightness of any neighboring point at which constructive interference occurs briefly then increases. The exact location of this point depends on the value of the phase jump.

As a result of decreasing coherence length, the mean brightnesses of all measuring points are equal. For very brief moments, there can be constructive interference at any point and a sequence of images with extremely short exposure times would show chaotically hopping points of light. As the coherence length increases, the dwell times at certain points become longer and longer, and the well-known interference pattern from regularly arranged bright points becomes more and more apparent. In the case of an infinitely large coherence length, a constant high brightness would be measured at some (regularly arranged) measuring points, the areas in between would be constantly unlit.

## Basics

Interference signal (3) as a function of the path length difference. (2) is the electric field strength of the interference.

The figure shows the effect of the coherence length on an interference signal. Curve (3) is the intensity of the interference signal as a function of the path length difference. In this representation, the coherence length is the width of the envelope  (1) at half the amplitude .

## Applications

Coherence lengths are used in different optical measuring methods: