Coherence

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

First lets start with what coherence/correlation is.

broadly the ability for light to constructively/destructively interfere with itself. 
Suppose we have a point in space denoted P, where two incident plane waves originating from sources S1 and S2 intersect.

We denote the total electric field intensity at this point as

Therefore, we say that the total intensity at that point is 

Expanding this out, we have 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\langle(\cdot E^*>=<(vec E_1+\vec E_2)\cdot (\vec E_1^*+\vec E_2^*)>
=<\rangle=\langle |E_1|^2 + |E_2|^2+2Re 2\text{Re}(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=\rangle = I_1+I_2+2Re<E_1\cdot2\text{Re}\langle E_1E_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)

...

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

\rangle

Evidently, the total intensity is the sum of the two intensities, plus a cross term. This is the "interference" term.
When no interference occurs, we say that the two beams are mutually incoherent.

We can now try to quantify this notion of coherence a bit more by focusing on the interference term. 

Suppose that the two 

There are two notions of coherence: Spatial and Temporal.

Temporal coherence asks: if I take a single point in space and look at the magnitude of the E field at t=0, do I have any information about what the E field will be at some later time t=tau? ie, does the E field at that point in space have a steady phase over time?

Spatial coherence asks: if I take two separate points in space, does knowing the E field at one point give any information about the E field at another point? ie, do two spatially separate points have the same phase relationship?

• Suppose we have a single point source S, which emits perfect monochromatic light with a random phase in short random bursts of time. This source is thus highly temporally incoherent because you check the E field at one point in time (where it is emitting perfect monochromatic light with a constant phase but for a very short amount of time), and then when you wait a bit of time the source has already changed up what phase it emits with so then you now have no idea what the E field at time t+tau will be.
• However, this source is still perfectly spatially coherent! At any point in time, you measure the E field at two transverse points (points that are the same distance from S). because its a single point source, you know they must have the exact same E field!

Spatial coherence has two aspects: transverse and longitudinal. this is only based off what your choice of those two points are.
We've already discussed transverse spatial coherence. This is what people typically mean when they say "spatial coherence". Essentially we ask how correlated the E field is between two points that are transverse to the direction of propogation.

Longitudinal spatial coherence asks how correlated the E field is between two points along the same direction of propogation. This is essentially just set by the coherence length of the source, which is actually determined by the coherence time of the source and thus is highly related to temporal coherence. For two points that are closer together than the coherence length (c*t_coherence), the E fields are highly correlated. For two points further than the coherence length, the E fields have no correlation

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.