Okay, good morning. Let's start. At the last lecture I didn't finish something I wanted to finish.

And that's something I find important. Because I spoke about the coherence.

And I said that you can define coherence as... you can define something like a coherence volume.

So if you fix a point in space, then around this point the field is coherent to the field of this point.

It's obvious, yes, some obvious argument.

And this point can be in the far field, then a point indicates some k vector, some direction.

And then in time also this point indicates some time, moment.

And then we define coherence time and coherence radius in terms of, of course, plain monochromatic waves, right?

Or time single points and space, k-space single points.

But there is a different definition of coherence.

Namely, if you write the first order correlation function g1 as a function of t1, t2, r1, r2.

And these are time and space arguments of the first order correlation function in the general way.

Let's consider the case where the field is coherent.

And let's reduce our consideration, for instance, to only time or only space arguments.

So, for instance, I consider only space.

Then it is g1 of r1 and r2.

You remember, hopefully, the definition that there are fields at points r1 and r2.

And then we introduced the normalized correlation function g1 as r1, r2.

As g1 at r1, r2 divided by square root of intensity at point r1, mean intensity, mean intensity at point r2.

And whenever this normalized correlation function was unity, it meant that the field is coherent at these two points.

Okay, so let's understand in what case this is really unity.

And it's rather obvious that it is unity when g1 as a function of these two arguments factorizes in two factors.

So it's a product of two factors.

First of them is equal to this intensity and the second is equal to this intensity.

So let's consider g1 is given by, and the, well, I can write like some f1 of r1 times f2 of r2.

So two functions, any two functions, one of them relating only to the first coordinate, the other relating to the second coordinate.

And immediately, because we know that g1 at r1, r1 should be equal to the intensity at point r1, mean intensity.

Right, that was as well the definition.

Then from this, you can immediately figure out that g1 in this case, in this case where the not normalized correlation function is a product of two factors like this, factorable.

Then you can write that g1 at r1, r2 is just, we write this product f1 of r1 times f2 of r2.

And then here, the intensity at point r1 will be just f1 at r1 times f2 at r2 squared, yeah, squared, yeah, square root of this.

I just substituted the definitions. And then from this, you find that the modulus of g1 of r1, r2 is unity.

Right, so here the intensity at point r1 is just g1 at point r1, so this is f1 at r1, f2 at r1, and for the other point you get the same.

So you get this expression, and then you get that the modulus of g1, is it clear?

Yeah, so assuming that the correlation function factors, you immediately get that the light is coherent.

Clear?

How do we get the squares?

We get the squares because g, because, okay, intensity at r1, mean value, is the same as g1 at r1, r1, right?

And then it is f1 at r1 times f2 at r1, right? And then the other two you get from the other intensity, yeah?

So if the correlation function factorizes like this, then the field is coherent everywhere.

And then, so this was the first point, and the second point, there exists in mathematics so-called Mercer's decomposition, or Mercer's theorem.

And it's related to another term, Schmidt decomposition, they differ slightly, but it's very similar.

So the mathematical statement is that if some function, and for us it's of course g1, r1, r2, is a good function,

so without singularities, without some big problems, and in physics functions are good usually, so more or less any function, g1, r1, r2, can be written as a sum of such products.

Not such a single product, but a sum of such products. And the sum looks like this, so there is a sum over index n, then there are some coefficients, alpha n.

And here, actually what stands here is the same function, f of r1, complex conjugated, times, and it has an index n.

And then function f of r2 with the same index, so the same function, so any correlation function can be decomposed into a series, into a sum, infinite sum in the general case.

So n takes values from 1 to infinity in the general case, so there is an infinite number of such terms, but each term is a factor, is a product,

so there is some function of r1 and some function of r2. And the problem is to find these functions, and these functions are called modes, coherent modes.

Or you can write it as the sum of n from 1 to infinity of gn of r1, r2, and each gn, with the weights, alpha n.