## Interference: Coherence length

For differences in path length greater than the coherence length, phase relations between two different rays become random. Here we illustrate the phenomenon with (oversimplified) cartoons, including the one at right.

### Coherence length: definition

Consider two different rays from a single source, which later combine. If their optical path lengths are exactly equal, they interfere constructively. If they differ by half a wavelength, the interfere destructively. If the differences in pathlength is sufficiently long, however, then there is no correlation between the phases of the rays. This effect defines coherence length: for difference in path length greater than the coherence length, phase relations between two different rays become random. To illustrate this, let's look at the cartoons below.

### Cartoon illustrations

These highly simplified cartoons show thin film interference. The first shows a non-reflective coating on glass: a thin layer whose refractive index n is less than that of glass, and whose thickness is λ/4n. (For more details see Non-reflective coatings.) First, consider the case where the difference in optical pathlength (here 2nt = λ/2) is rather less than the coherence length, which is 2λ in the cartoon. (For light produced from hot objects, the coherence length can be of this order.)
 Optical path length difference shorter than the coherence length

In the cartoon above, we see destructive interference, as expected. The two sine wave segments overlap over most of their length, and they are π out of phase.

 Optical path length difference shorter than the coherence length

In the second cartoon, the difference in optical pathlength is 2nt = /2, which would produce destructive interference for light from a highly coherent source. However, this is longer than the coherence length of the light in our cartoon, which is again 2λ. Here there is no overlap and so no interference.

In practice, there are usually many photons simultaneously. However, those that have travelled further need not have a correlated phase, because they were emitted by a different event that occurred earlier.

### The photon question

Remember the question in the multimedia tutorials Interference. Here's one way of putting it: Consider photons striking a thin film, as in our animation, and producing destructive interference. Suppose that one photon is reflected at the first interface. How can it later be cancelled out by a photon that hasn't even reached the second interface yet?

The short answer is this: When we say 'striking' or 'reaching' an interface, and that one has reflected before the other, we imply that we can define the times of arrival very precisely, which means a very a very short coherence length and so little interference.

Alternatively: Talking about particles implies localisation in space. Via the uncertainty principle, this means that we don't know the wavelength well, so interference will not be strong. On the other hand, if we specify the wavelength precisely to ensure strong interference, then we can't talk of one photon being reflected much before the other.