 Now, we start next part of our discussion and what we want to talk about really is we have been using and we have demonstrated when we went to the lab and we talked about it we have been using non-linear optics quite a bit. In the last module we had talked about your amplified laser the laser that we have the next step is optical parametric amplifier. Now in order to understand optical parametric amplification first of all we need to build a little bit of basics of non-linear optics. And as has been the central theme of this course we are not going to try to derive everything perhaps but we will try to understand how things work. So what we learn today is this when we showed this femtosecond optical getting experiment we had shown that red light comes false on some non-linear crystal and then when we turn the crystal by using a micrometer screw we see that second harmonic is generated to a greater extent. So today in this module we are going to learn why that happens why is it that you turn the angle of a crystal and you get good second harmonic while you do not get it earlier to do that we have to I am sure many of us know already about non-linear optical phenomena. So we will discuss the basics and let us see how far we can get. So first of all we have talked about this earlier and I am sure everybody has learned this in other courses. The model of interaction of light with matter is that light induces a polarization in the medium electric the electric field component of light induces a dipole moment electrical dipole moment and that is what interacts with the electric field to give you the radiation matter interaction which is what we are talking about. Now if you have ordinary light then you have linear response meaning the polarization is directly proportional to the amplitude of the electric field E. So you can write it in a very simplistic manner as P equal to chi first order that 1 in the superscript within bracket means first order it is not chi to the power 1 it is not chi 1 it is chi first order. So P equal to chi first order E where P is polarization first order chi is the electric susceptibility of first order and E is the amplitude of the electric field but then there is actually more to this than what we have written and someone tell me what more is there what is it that we are actually hiding in this simplistic expression. No that even before going to square cube term even in linear response suppose the response is linear I am saying even then in that case this equation that we have written is a tad too simplistic the situation is more complicated than that ok Hint Raman spectroscopy you have learnt this in Raman spectroscopy yes not dipole moment susceptibility so let us say alpha instead of chi first order I will write alpha polarizability let us say if I write polarizability does that ring a bell think of what you learnt in Raman spectroscopy this expression is actually more complicated than what it is ok electric field is it a scalar or vector the vector field right what about polarization scalar or vector yes so now Prajith has remembered the thing is this it is not necessary that if I apply an electric field along x direction polarization will be along x direction only electric field along x direction can produce polarization in other directions as well so if you write P the three components of P P X P Y P Z let us say we write it as a column vector a column matrix that will be equal to column matrix E X E Y E Z left multiplied by a square matrix right and the components will be something like alpha xx alpha xy alpha xz and so on and so forth why is it that if I apply electric field along a particular direction polarization is produced in other directions as well yes I am applying in one direction so to answer this question the easiest approach is to think like a baby like a kid think of simple analogies it is easiest to understand that way ok what is the meaning of polarization production of dipole moment what is actually happening to the molecule electron right so electron is it particle or wave or what right so electron cloud and the electron cloud is distributed all over the molecule so when you polarize what you do essentially is that you distort the electron cloud is that right so let us say we are spherical electron cloud over the molecule you distort it so that in one direction the concentration of electron cloud is more in the other direction it is less ok I like to think of it as a balloon this entire electron cloud is like the air contained inside a balloon now think of holding a balloon and pressing like this well you can pull also but then it is easier to press so this is the field you are applying and the balloon will get depressed in this direction but is it not elongated in the perpendicular directions X and Y yeah so that is a rough analogy that explains why when you apply an electric field in a particular direction distortion takes place not only in that direction but also in perpendicular directions all directions that is why it is actually a tensor for our purpose for now it is ok if you write like this right next we come to what Shoradip was saying this expression that we have written in the simple form or in matrix form this is valid when we have ordinary light light coming out from the electric bulb where the field is weak what happens when we use a laser a laser is associated with significantly higher electric fields right then the response is no longer linear then you have something like this your nonlinear response P is equal to chi first order E plus chi second order E square plus chi third order E cube and so on and so forth and the reason why you see this for high values of E and not for low values of E is that first order chi is much larger usually then second order chi if it is provided the second order chi is nonzero we will come to that second order chi is generally much larger than third order chi and so on and so forth that is why unless this E square term or in some cases E cube term can take over unless the value of E is sufficiently large you do not see the higher order terms ok but when you deal with lasers you can get at least the second order term sometimes you can get the third order or fourth order term as well we will see what are the conditions that are required for this higher order terms to be involved ok for now let us focus on the second order chi it is called the second order nonlinear susceptibility ok so this is the beginning of our discussion of multi photon processes 2 photon process of course is the simplest multi photon process one can think of and the second order nonlinear susceptibility is responsible for bringing about 2 photon processes so once again let us remind ourselves of what 2 photon processes are so whatever we are discussing now I believe is known to all of us is just the capitulation what is the meaning of 2 photon absorption let us say we have these 2 energy levels separated by an energy equivalent to frequency of 2 nu or nu 1 plus nu nu 2 depending on whatever we are going to write so to start with let us say the energy gap is equivalent to frequency of 2 nu and let us say I excite with laser pulse of or not necessary the pulse laser light of frequency nu what will happen the photon does not have enough energy to cause a transition to the next available stationary state right but once again cutting a lot of mathematics out and using a little bit of childish analogy the molecule does not know right the molecule does not really the molecule has not studied quantum mechanics so it does not know when the light comes that it is not supposed to absorb it so what happens is the distortion of the electron cloud starts anyway and then it reaches some virtual state what is the virtual state what is the stationary state what is the virtual state what is the stationary state CH 107 stationary state is a state in whose energy or whose psi psi star we can say is independent of time ok so these are states associated with finite life time your molecule can stay there for some time and virtual states are anything in between you can have an infinite number of virtual states between any two stationary states and the way you generate these virtual states is by taking linear combination of stationary states and the point to remember is that virtual states are associated with lifetime of 0 the molecule cannot stay in the virtual state for any amount of time so the moment it is there is going to come down so if your intensity of light is not too much then you will not see any absorption so you can think like this the electron cloud starts getting distorted this is the shape of electron cloud in the ground state this is the shape in the excited state it starts getting distorted runs out of gas midway and comes back however if you use intense laser light then what happens is at the same moment when the molecule or let us say the system is promoted to the virtual state there are many other photons around so a second photon can cause promotion to a second to the stationary state and if you carefully choose the frequency of the photons you use so that their energy is exactly half of the energy gap between two stationary states then you can have what is called two photon absorption okay so this is a very very simplistic hand waving way of saying what two photon absorption is and the case we have discussed is the simpler of the two cases of two photon absorption where it is but both the photons are the same color same energy same frequency regenerative photon absorption that need not be the case always sometimes what happens is that the energy gap might be equal to the sum of the frequencies of two different kinds of photons that you have at your disposal in that case first photon let us say new one takes the system to the virtual level and at that instead a large concentration of new two photons are available the molecule is promoted to the next stationary state of course all this is just wishful thinking we are writing like this to understand there is no reason why new two will not be absorbed first we do not know actually which have been absorbed first which has been absorbed first the other analogy I like is that it is sort of like two water droplets joining up together and forming a larger droplets so two energy of two photons are sort of getting added up to give you the promotion that you require okay so much for two photon absorption but then when you absorption when absorption takes place the energy of the system goes up so anything that goes up has to come down so let us talk about some frequency generation or in some cases two photon excited fluorescence let us take the second case where the energy gap is equal to new one plus new two non-degenerate to photon absorption now and let us say that the energies of new one and new two are such that new one plus new two is not equivalent to any energy gap between two stationary states in that case even after new two is absorbed the system is going to be some virtual state right what is the lifetime of a virtual state 0 so the moment the system has reached that virtual state it cannot stay there immediately it has to come down and it comes down with emission of the light that is absorbed but then there is no memory the system does not remember that I had absorbed new one first and new two second or the other way around there is no memory so when the emission takes place the emission is of a single photon of frequency new one plus new two this is how some frequency generation can take place now think of a case where this higher energy state is not a virtual state but rather a stationary state then what will happen stationary state is it associated with some non-zero lifetime or it is a lifetime 0 generally lifetime is not 0 so when the molecule gets excited to a stationary state and then the emission takes place from there what is it called fluorescence in the simplest case scenario yeah same spin multiplicity so this is something that is used especially in microscopy where you excite where you do a two photon excitation and you look at the fluorescence the problem is this let us say I want to look at some protein under a microscope what are the the fluorophoresin protein generally tryptophan is the major one anything else tyrosine anything else phenylalanine this is something that is perhaps the least popular there is a school that has tried to promote tyrosine as a good fluorescent proven all but generally people follow tryptophan emission ok if I want to follow tryptophan emission where do I excite 295 but tryptophan absorption maximum is 280 nanometer why do I excite at 295 nanometer so as to avoid exciting phenylalanine and tyrosine ok so that is just a recap not really anything to do with the present discussion now the problem is this I want to look at a protein that is inside a cell under a microscope and I want to follow it by tryptophan emission then I have to excite by 295 nanometer light two issues there first of all unless you buy a significantly more expensive UV microscope regular microscope optics are all glass so 295 light won't even get through secondly even if you use a UV microscope you are looking at cells which is a turbine medium they have to scatter light scattering efficiency how does it fall off with lambda lambda to the power what lambda to the power 4 so these are things we cannot forget we have to remember so UV light is scattered big time you won't see anything so it is much better in that case to not use 295 nanometer light but rather use some laser which gives you 295 into 2 how much is 295 into 2 302 into 2 is easier 600 600 minus 10 590 so 590 nanometer laser if you can use intense 590 nanometer laser then you can have a 2 photon absorption to a stationary state right and then you can look at the emission of the protein provided that emission is not scattered provided that emission I mean provided your optics that can look at that emission that is a different issue 2 photon excited fluorescence is very popular in biological systems because the problem is this if you use wavelength that is too short it will get scattered if you use wavelength that is too long it will be absorbed by water so typically when wants to do microscopy one wants to work in the therapeutic range of 650 nanometer to 900 nanometer but then if you excited 700 nanometer where will it emit it will emit beyond 900 2 photon excitation followed by fluorescence is very popular technique especially in microscopy okay but here coming back to our present discussion what we have seen here is how you can get some frequency generation so in some frequency generation there is no delay light is absorbed instantaneously light comes out in instantaneously you do not have to wait any time so if nu 1 plus nu 2 is the case we call it some frequency generation and in the special case when nu 1 and nu 2 are equal to each other then we get second harmonic generation that is what we do in inside our millennial laser that is what we do for in the fog instrument and in any other places okay. So now with that simplistic background build let us go a little further into understanding second order non-linear phenomena okay so once again we are following this our usual textbook Helena now think of what you have in the entrance side of the fog spectrometer you have light from a laser here we have written with frequency of omega it falls on some non-linear crystal we will talk about what optic axis is in a little while and then you get omega and 2 omega out and then you use a long pass filter to get 2 omega out and block omega or do something in our case what we do is we transmit omega and we reflect 2 omega fine what you have to use a dichroic here once again we proceed in the simple with the simple description of the system the electric field of light is a function of position as well as time that R that we have written anything that is in bold is actually a vector electric field is a function of position and time and it is given by E 0 multiplied by cos of k 1 R – omega 1 t what is omega 1 there is no retry to omega 1 here actually I could have written omega but later on I have talked about omega 2 and all that is why here omega 1 is there what is omega the answer is there in the projection omega is the angular frequency of light incident light what will be angular frequency of the second harmonic 2 omega remember what is R R is in bold so it is a vector what is that R small R there are very few things for which you use R and a vector at that what is R just tell you what do you use smaller for distance right and R vector does that ring a bell distance from where is the position vector distance from the origin you have some point x y z you draw an arrow from the origin to that point that arrow denotes the position vector R is that same position vector nothing else what is k 1 k 1 once again is a vector it is called the k vector is there another name that you know wave vector so what is k that is something that we should say and then we can stop this module go to the next one we will come back to that but first of all suppose this is E and for this expression of E I want to write the expression for this second order polarization what will it be just a second order term not the first order term remember what it was p equal to chi first order multiplied by E plus chi second order multiplied by E square so what will be the second order term for polarization chi second order multiplied by well E square what is E square here E is equal to E 0 cos k 1 R minus omega t omega 1 t so what is E square E 0 square that is a good beginning then cos square k 1 R minus omega 1 t it is very simple so as I said many times in many different for I never ask difficult question cos square term but now I want to ask a question let us see if you remember our high school high secondary trigonometry how do we write cos square theta cos square theta is equal to well in terms of cos since you are writing cos we will stick to cos 1 plus cos 2 theta divided by 2 right 1 plus cos 2 theta divided by 2 what is theta here k 1 R minus omega 1 t right so can you work out the expression E 0 square do it in terms of theta E 0 square cos square theta what does it become in terms of cos 2 theta E 0 square cos cos square theta if I want to write it in terms of cos of 2 theta cosine of 2 theta what do I get yeah E 0 square by 2 plus as I have taken 2 common plus there is a half as well and do not forget do not forget second order chi I might have forgotten to say it okay so that will come later fine E 0 square by 2 plus cos 2 theta divided by 2 okay that is E square now to get second order term for polarization this term has to be multiplied this factor has to be multiplied by second order chi second order susceptibility so this is the answer second order term for polarization is given by second order non-linear susceptibility multiplied by E 0 square by 2 plus half second order non-linear susceptibility multiplied by E 0 square cos 2 theta where theta is k 1 R minus omega t okay what is the frequency of this associated with the second term what is the frequency angular frequency associated with this second term this one 2 omega 1 that is right so look at the first term the first term is a constant polarization is it not there is nothing in t there it is just second order non-linear susceptibility multiplied by E 0 square by 2 E 0 is a constant of time constant of space constant of everything right so this is called constant polarization or DC effect this part is used to measure the power second one is actually the second harmonic term there you have a polarization that is modulated by a frequency of 2 omega 1 so this is the term that is responsible for second harmonic generation okay then we take a break here and in the next module we start right here