Okay, let's start. Today is our second lecture on quantum optics, statistical and quantum

optics, and I want to tell at the very beginning that this lecture as well as most of my next

lectures will be different from what was last year, well slightly different because the

subject of statistical optics didn't change during the last year and the subject of quantum

optics also didn't change, but parts of my lectures of last year I replaced and this

will be the case today because I decided to remove for instance Schmitt modes and coherent

modes and to add optical coherence tomography instead. So today's lecture will be about

correlation function measurement interferometers and a little bit about photon number per

mode. And this lecture I am not sure that I will put notes on the website because I

haven't prepared them yet, but of course everything is recorded. Now last time we

introduced the correlation function g1 in the general case of time, a function of time

and space. So I will write the most general form t1 r1 t2 r2 and this is the correlation

of the fields at points, time points t1 t2, space points r1 r2 and let me remind you the

definition so this is the negative frequency field at t1 r1 and positive frequency field

at t2 r2 average. And the average will be now statistical average over the time for

instance or the ensemble and we will assume that the processes are, our processes are

ergodic, it means that we can average it over time or we can average it over realization.

And in the stationary case it will be, it will only depend on the times difference t2

minus t1 and in the homogeneous case it will depend only r2 minus r1 so it will be g1 of

tau is equal to t2 minus t1 and rho equals r2 minus r1. So the question is how to measure

this correlation function because at the last lecture we learned an important thing that

it is Fourier transform of the spectrum, the Wiener-Hinchen theorem and it means that if

we want to know the spectrum and the spectrum is for instance too narrow for us to measure

then we can measure this correlation function but how to do it? Obviously one has to have

these two fields separated by some time and some space and then find the correlators,

find the average of their product. And we'll start with the time measurement so time correlation

function which means that we will consider just g1 of tau and we will ignore the space

coordinate, we will consider just the same point, the same spatial point, we'll have

r1 and r2 coinciding and then the easiest way to do it is to use an interferometer.

So the strategy is as follows, we send some field here E of t and then we split, we put

a plate that is a glass plate usually that is semi-transparent so it transmits some part

of this field and reflects some part of this field. It has transmission coefficient and

reflection coefficient and for the most convenient way is to make them equal and equal to 1 over

square root of 2. These are transmission and reflection with respect to the field so with

respect to intensity it will be one half the square of this thing so it's called a balanced

or 50% beam splitter, 50% beam splitter. Later we will see that actually these coefficients

do not have to be equal, they can be unequal but we will discuss what are the consequences

of them being unequal. So we'll split this field on a beam splitter and then we reflect

back by this mirror the reflected field and then we reflect back also the transmitted

field and then use the first one that was first reflected then is transmitted and the

other one the other way around. So here we have contributions from both paths and we

have the option to move one of the mirrors by some micrometer or some stage and here

we put a detector and we measure the intensity and we look at the interference. So this thing

is called Michelson interferometer. Let's see what happens at the output of the interferometer,

the field at the output. So this is the intensity at the output and this is just the field at

the output minus the negative frequency part of t times e plus out of t. This is the intensity

as a function of time, exactly how we wrote in the last lecture. And so we have to find

this field at the output and of course this field at the output has contributions from

fields that were reflected and then transmitted and then from the field that was transmitted

and then reflected and we write that the output field of t and let me write the positive frequency