Coherence

In general coherence is the same thing as correlation, but applied to electromagnetic waves.

First lets start with what coherence/correlation is.

Lets say that we have two sources S1 and S2 emitting EM waves, and they combine at some point P.

The intensity at this point is thus

I=<E\cdot E^*>=<(E_1+E_2)\cdot (E_1^*+E_2^*)>
=<|E_1|^2+|E_2|^2+2Re(E_1\cdot E_2^*)>

Where <> denotes a time average defined as

<f>=\lim_{T\rightarrow\infty}\frac{1}{T}\int_0^Tf(t)dt

This reduces the intensity down to

I=I_1+I_2+2Re<E_1\cdot E_2^*>

For a typical interference experiment, we take a single source and then route the light through two different paths and then recombine at a point P.

suppose that the light E_2 takes an additional time tau to get to P compared to E_1.

We define the "mutual coherence function" as follows

\Gamma_{12}(\tau)=<E_1(t),E_2^*(t+\tau)>

And then we use this mutual coherence function to repackage the cross term in the intensity

I=I_1+I_2+2Re\Gamma_{12}(\tau)

Now we define the "degree of partial coherence"

\gamma_{12}(\tau)=\frac{\Gamma_{12}(\tau)}{\sqrt{\Gamma_{11}(0)\Gamma_{22}(0)}}=\frac{\Gamma_{12}(\tau)}{\sqrt{I_1I_2}

where we essentially normalize the mutual coherence function by I1 and I2

We then again update the intensity formula

I=I_1+I_2+2\sqrt{I_1I_2}Re\gamma_{12}(\tau)

Okay so now we know how to find the measured intensity at some point if we have two sources coming in that have travelled different optical path lengths.

It seems that it depends on the value of gamma; let E1=E2=E. When gamma=0, I=2I. when gamma=1, I=4I.

We say that when gamma=1, we have complete coherence and when gamma=0, we have complete incoherence.