Thank you very much. So let's start our last lecture this year and the plan of this lecture

is I will tell you about how nonlinear optics provides effects to generate non-classical

light. And in the first part I will consider just some hand waving examples. So I will

write just a few equations and explain physically what happens with the second harmonic generation

and Kerr effect. But the second part will be more rigorous. I will derive the Hamiltonian

for paramedic down conversion and I will describe the non-classical features that appear in

this kind of process. So let's start. As usual the home task I will, the solution to the

home task I will explain in the end. So we start with nonlinear optics and as we already

discussed in the general case the polarization is not linear in the field but it has, it

can be written as an expansion. So the part epsilon naught and then there is chi 1 e,

e is the field, thank you Gaetano, p is the polarization and in addition to this linear

term there is second order nonlinear term and I omit here the tensors. I assume that

everything is just isotropic so the field, the field squared and then chi 3 e cubed and

so on. I will not continue further because all we need today is this term and this term.

And as we wrote before the creation of nonlinear polarization leads to the generation of field

again so what happens is there is a field here, you find the nonlinear polarization

and this polarization in its term generates the field at probably other frequencies and

this process is governed by the Helmholtz equation. So let's first consider non-zero

chi 2, chi 2 is non-zero so this is the case of materials without center of symmetry, crystals

of course and then we are just interested in the second order polarization which is

proportional to epsilon naught, chi 2 or not proportional but just equal to epsilon naught,

chi 2 and square of the field. And then if we assume that the field is a harmonic field,

not even just a harmonic field at frequency omega which means that this is e naught e

to the power minus i omega t plus ik z. I consider a wave propagating along the z direction

and then there will be complex conjugated so this is actually the positive frequency

part of the field and this is the negative frequency part of the field then by taking

the square we will find that second order polarization will be proportional to there

will be a term e naught squared e minus 2i omega t plus ik 2i k z. So this is the square

then there will be the result of the negative frequency field e naught complex conjugated

squared e 2i omega t minus 2i k z and there will be of course the cross term which will

take the form. Note that I am writing everything classically these are classical fields so

the order doesn't matter there will be here e naught squared and the exponentials with

a factor of 2 of course. The exponentials cancel. So here we see immediately two different

effects the appearance of a constant field while this constant polarization will create

a constant field this is called optical rectification and we will not consider it in this lecture

but the first term and of course because this is the sort of positive frequency part and

this is negative frequency part this term generates this term describes generation of

the second harmonic second harmonic generation. So we see because we see that double frequency

appears here and also note that polarization is proportional to the square of the field

then we have to write the Heffield's equation to describe how the polarization in its turn

creates the field and I will just write it down because we derived it in one of the lectures

nabla squared e and this is already the field due to the nonlinear polarization so I will

put index 2 here minus n squared over c squared n is the refractive index and then second

derivative of this field two dots mean second derivative as usual and this should be equal

to 4 pi over epsilon naught c squared and this p2 again second derivative. So we have

to take second derivative of this polarization and this will substitute into this equation

but it is already clear that because this equation is linear with respect to the field

then the well in the left part there is linearly field and in the right hand part there is

this p2 and p2 is proportional to squared field in the incident beam and is oscillating

at twice the omega so it is clear that e2 will be proportional to e naught squared and