# What is Correlation length ?

1. Jan 21, 2014

### Ariel

What is "Correlation length"?

Hi everyone,

I want to know meaning of "correlation length".

Correlation function is usually used in statistics, so I think correlation length also similar meaning.

But some paper used correlation length to explain roughness factor.

Thanks a lot to read this message. :)

2. Jan 21, 2014

### Cthugha

Consider a sample with a surface that is not perfectly flat, but somewhat rough. If you want to analyze the surface roughness, you can just measure the thickness of your sample at many positions, say every micrometer. Now, you have the mean sample thickness and the deviation from the mean for every point on the surface where you did a measurement. For a real surface, the deviation from the mean will not change randomly on every micrometer, but there will be some extended "hillls" and "valleys" on your sample. So, if you check the deviation from the mean at one sample position and have a look at the deviation at an adjacent position, it is very likely that both will be similar, while it is very unlikely that one is a large positive deviation and the other is a large negative deviation. If you have a look at the deviation from the mean thickness at one end of the sample and the deviation from the mean at the other end of the sample, the two values should be completely uncorrelated. They may be large, small or zero - completely independent of each other. Now somewhere in between , there must be a typical length scale on which the deviations from the mean stop being similar. That is the correlation length. It gives you some information on how large these "hills" and "valleys" on your sample typically are.

Needless to say, you can apply the same principle to other quantities as well, with applications ranging from surface roughness to the optical coherence of light fields.

3. Jan 21, 2014

### Ariel

If correlation length is short it means surface fluctuation is strong, so rough surface, otherwise fluctuation is week, so flat surface. Is my understanding correct?

4. Jan 21, 2014

### Cthugha

In practice, rough surfaces usually have short correlation lengths, while smooth surfaces have a long coherence length. However, that is only a rule of thumb and not necessarily always true. For a detailed analysis one typically also takes the RMS of the surface height into account. This is more or less just the standard deviation of your measured thicknesses. This quantity gives you the amount of roughness you have and the correlation length gives you the typical length scale over which it decays.

However, as you said, in most real cases short correlation length indeed also means strong fluctuations.

5. Jan 21, 2014

### M Quack

Think of a sine modulated surface. The amplitude of the sine gives you the roughness, and the spatial frequency the correlation length.

In practice you will have the superposition of many sine waves...

6. Jan 24, 2014

### Claude Bile

Correlation in this context means "given the height of point A, how well do I know the height of point B?".

Given the height at A, the correlation length is the maximum distance B is from A, that yields a reasonable* estimate of the height at B.

*definition of reasonable may vary.

Correlation length is linked to the power spectrum of the surface noise. Shorter correlation length means higher spatial frequencies dominate (steeper hills etc).

Correlation length is distinct from the amplitude (amount) of surface roughness.

Claude.