The coherence length, usually applied to light waves, gives the distance over which a wave can interfere with itself. For example, imagine a source of light being split into two beams. The beams are then taken on different paths and later recombined. If the difference between the path lengths is less than the coherence length then the beams will interfere with each other and an interference pattern can be observed. If the difference is greater than the coherence length then no interference will be observed.

From this it can be seen that the concept of coherence length is extremely important in holography as it is the process of splitting and recombining a beam to produce and interference pattern that makes a hologram. This is why laser light is usually used in holography, as lasers have extremely long coherence lengths, in the order of a few metres is possible. Compare this with the few centimetres possible with a sodium lamp.

The coherence length depends on the purity of the light source. A laser emitting at a single wavelength, λ, will have a very narrow spread in wavelengths, Δλ. Traditional light sources, such as sodium lamps, whilst only emitting "one" frequency will have a larger spread in wavelength, Δλ, than the laser.

Formally, coherence length is defined as

L = λ² / nΔλ

where λ and Δλ are defined as above and n is the refractive index of the medium in which the wave is travelling.1

To fully understand the origins of coherence length you have to appreciate that for most light sources the light is not emitted in a continuous wave, but in discrete bundles, or wave packets. A wave packet is released from an atom as one of its electrons drops from an excited level to a lower energy level, releasing the difference in energy as a photon, or wave packet:

```
Y
..     ;MU
KM     ZJH
.KN     m,N
2ZU;    N E:    2m.
m;J2    N VJ    HJV
.J     N ,m    H JZ    N K
NH,    N .H   ,m .H   ,E H
JU:K    H  N   JV  N   JZ Z2    77
,     N. M   JV  H   ZJ  H   U7 JV   JZJU
,,,.   .;77.   2V2K.  .E  m;  YL  K:  U:  N   K.  N   N  ZY   2YY7    ,,,:
777J2222;.:LYYV;  :H  m;  2Y  U;  m,  E   H.  N   N  .H   E;7V7  ,LYY2777J222J
7K.N   .K  N   JZ  H   ZJ  N   U7 m7    2;
7Y:    N  N   7U  N   LL ,E   :K N
E, H   ,m  H   JV LY    MYY
7VLY   .H ,E    H U7    ;2
.KK:    N 7U    N K:
ZK     H ZJ    N N
m,H.    H.K
LYN     VHL
.NN      ;
U7

```

Each of these wave packets has absolutely no relationship to any of the others, so there is no fixed phase relationship between them. If you take two wave packets arriving at the same point, they may interfere constructively or they may interfere destructively. Take two other packets at the same point and they may do the complete opposite. So, on average after many hundreds of millions of these wavepackets you get no fixed interference pattern. This is the reason we only see laser speckle with laser illumination, not normal, white light.

In lasers, because the radiation emitted is stimulated by the other photons present, each wave packet is emitted in phase with the rest. This has the effect of making one big long wave packet, leading to a large coherence length.

1. http://www.its.bldrdoc.gov/fs-1037/dir-008/_1055.htm

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