So I believe we start. If someone, some of you made the task that I gave you, you can

give it to me after the lecture, but I will not discuss this problem because it will be

Cameron who will do it at the problem class to save the time during the lecture. Yeah,

yeah, after the lecture you can give me. Very good. So at the last lecture we introduced

the main concepts of statistical optics, in particular random field, random intensity

and the correlation function. Today we'll continue further on this first order correlation

function and I will discuss the ways to measure it. So as I introduced g1, in the general

case I will put all arguments t1, t2, r1, r2, meaning that this is the correlation function

between two temporal moments and two spatial points and I will remind you that this is

mean value of e minus at t1, r1 and of course r is a vector in the general case, right?

e plus at t2, r2.

I have a question.

Yeah?

So we put a semicolon between the time and the space.

We put a what?

A semicolon and the space.

You mean you want it to be written like this or?

I don't think it matters.

But I'm here, this is what my question is.

Doesn't matter I think. Anyway you take a field at time 1, t1 at coordinate r1 and you

take a field at t2, r2 and this is negative frequency field, this is positive frequency

field, I explain what it means and then you take this product and average. And in particular

cases you can have r1 equal to r2 or you can have t1 equal to t2, depending on this you

will only study temporal correlations or spatial correlations and we explained at the last

lecture, I explained at the last lecture that due to stationarity if the process is stationary

then g1 depends only on the difference of the times t1 and t2 and if the process is

uniform in space it will depend only on the difference of r1 and r2.

Oh I see some non-students here.

So I can write g1 as a function of t1 minus t2 which will denote a tau and if you want

I put a semicolon here, r1 minus r2 and I will denote it as for instance rho. In the

case of stationarity and uniformity. So if the field is stationary and uniform in space.

And then one thing I wanted to add to what I told you at the last lecture is that you

can normalize this correlation function. It's convenient to normalize it to make it, well

if you want dimensionless, by introducing g1 small again of these same arguments t1,

t2, r1, r2. By definition it's this non-normalized correlation function g1 of t1, t2, r1 of r2

and divided by square root of you can write here d1, t1, t1, r1, r1, d1, t2, t2, r2, r2.

So this is the strict definition but of course I told you at the last lecture that this g1

is just intensity, mean intensity at t1, r1. So it's the same as t1, t1, t2, r1, r2. Square

root of mean intensity at t1, r1, mean intensity at t2, r2. Or in the stationary case and uniform

or homogeneous, I'm not sure which word is the best, so this is just g1 divided by mean

intensity. Because mean intensity doesn't depend on in the stationary and uniform or

homogeneous case it doesn't depend neither on time nor on space. So this correlation

function is called normalized first order correlation function and it's very important

because it characterizes the coherence. And at the last lecture let me remind you that

we derived the theorem, the Wiener-Hinschen theorem that sets a very important relation

that g1, I will write not normalized correlation function, is the Fourier transform. Let's

write g1 of tau. In this case I consider the stationary case and the correlation function

depends only on the difference of the time arguments t1 minus t2, tau, is the Fourier

transform of the spectral density. So we derive the relation that this is the integral in

omega and here there is this, sorry, s of omega, which is the spectral density, the