 Let us complete the discussion that we have been performing over the last two modules. We are discussing second order nonlinear phenomena and we have said without deriving that intensity of the second harmonic light is proportional to square of second order nonlinear susceptibility length of the crystal and something like sin square theta by theta square where theta involves delta k and I0 square of omega. And hence we have said that in order to get favorable conditions for getting second harmonic generation is you should use a high intensity of fundamental light you should use as much length of the nonlinear material as you can without messing up things. You should have high I mean not you the medium should have a high second order nonlinear susceptibility and finally delta k has to be equal to 0 that is what is going to that is what we are going to continue upon delta k has to be equal to 0. What is k is a vector associated with the k vector what is the meaning of delta k equal to 0 momentum is conserved in the process of second harmonic generation total momentum of the combining photons is equal to the momentum of the second harmonic photon okay. So from here the discussion we are performing is we want to know how it is possible to get second harmonic because one problem we had faced already is that the refractive index of 2 omega has to be equal to refractive index of omega and we have said in the earlier module that it is impossible to do it unless you have a birefringent crystal. So what we are going to learn in this module is how is it that birefringence helps and we have already talked about birefringence but let us recap very quickly birefringence means this you have a crystal which has an optic axis once again we will tell you we will discuss what optic axis is when light falls on it it gets divided into 2 parts one of the rays that passes through is ordinary ray and the other one is extraordinary ray. In the ordinary ray polarization is the same as the polarization of incident light in extraordinary ray polarization is perpendicular to that of incident light and then the most important thing is that for ordinary ray the refractive index of the material please understand the refractive index of the material throughout is the same no matter what the angle of incident is incidences however for the extraordinary ray refractive index depends on the angle of incidence. So now what we will do is we will try to drop polar plots with only one circle or oval shape will not draw contours or anything we will try to drop polar plots of this refractive index for ore as well as e-ray and we will do that for 2 different kinds of crystals as you will see. So let us say this is the optic axis what is the optic axis we will come to it wait a little longer and let us say this is the angle of incidence okay if optic axis is z and I call this angle between optic axis and the incident ray theta does that remind you of something optic axis I am calling z and I am calling the angle between the optic axis and this incident ray to be theta does that remind you of something r theta phi so it is well I have already said it is theta right now position vector is done this theta theta azimuthal angle okay azimuthal angle that can be different right there is no special significance of making this theta small and making this theta large we are always trying to say is in principle theta can be anything from say 0 to 180 or even 0 to 360 in this case. Now suppose for every value of theta of pens this is z this is the incident ray this angle is theta so for every value of theta I am going to plot the refractive index once for the ore once for the e-ray once for the ordinary ray once for the extraordinary ray can you tell me what shape I will get for the ordinary ray for ordinary ray if I plot for this angle theta 1 I plot n o therefore this angle I will plot n o it will be the same isn't it because remember for an ordinary ray refractive index is not dependent on the incident ray incident angle right so I will get a circle yeah and it is a polar plot so don't so I will get a circle like this this is a circle I get for n o where o stands for ore ordinary ray if I want to make a similar plot for the extraordinary ray what will it look like for an extraordinary ray remember n depends on the angle of incidence so n e if I call the refractive index of the extraordinary ray if I denote it as n e then it is going to have theta dependence it is not going to be a circle what will it be will it be a square will it be a triangle yeah what should it be it will be a distorted circle so it will be an ellipse right because the point is this let us say this is the this is theta and let us say this is also this is also theta this angle and this angle are same there is no reason why n should be any different for this position and this position right they will be same but the moment you change it will change in a different way so I am going to have an ellipse but there are two cases that can be there and this is a good time to learn what an optic axis is an optic axis is the direction in which if the incident ray propagates in the direction of the optic axis then refractive index of ore and e ray are equal to each other okay no separation between ore and e ray if incidence is along the optic axis that is what optic axis is are we clear yeah have you understood see n o remains the same any does not it changes but at some angle of incidence n o and any are equal that is a property that is there okay so that line that you can get in that direction that defines the optical act optic axis right so that is your reference so what I am saying is with respect to optic axis we are defining theta then theta equal to 0 then n o equal to any when theta equal to 180 then also n o equal to any okay now with that I can draw two different ellipses see what I am saying is at this point and at this point any and n o have to be the same but what happens in the middle any can always be smaller than n o or any can always be larger than n o if any is always smaller than n o then I get an ellipse like this okay any is shown in dashed lines and in the other case if and at this moment I realize that and this point my animation could have been better the other case is for theta equal to 0 n o and n o is equal to any but after that any is always more than n o then this will be the case agreed and this one the first one n o is greater than equal to any the second one n o is less than equal to any when is n o equal to any in both the cases when is n o equal to any yes when the incidence is along optic axis or in other words theta equal to 0 or 180 degrees have we understood this diagram yeah now let us get done with the definitions in the first case where n o is greater than equal to any for those crystals you call them negative crystals and of course if this is negative then the other one crystals in which n o is less than equal to any those crystals are called positive crystals alright now here let us say I have drawn this n o and any for the fundamental omega 1 now I will want to draw n o and any for 2 omega but before going further are we all clear have you understood this part have you understood what is going on because the next we have introduced one phenomenon already on this slide now after this we are going to introduce another phenomenon and they will add up to provide the condition for angle tuning for face matching so it is important that we are all on the same page at this point is there any question is there any doubt everybody has understood everything and I go ahead okay now I want to draw the surfaces for n o and any for 2 omega okay and let us say I will draw it for the negative crystal you can draw it for the positive crystal yourself and it is not very difficult also once we do it slowly so first of all what will happen will a negative crystal become a positive crystal for the second harmonic or will it remain a negative crystal it will remain a negative crystal so you get similar kind of a diagram yes next question is well since n o is a circle it is easier to talk about n o is n o going to be the same or different for 2 omega generally it is going to be different we have already said that right so is it is the circle going to be smaller or larger for 2 omega is the refractive index larger or smaller than that for omega it is smaller always smaller so what I will do now is I am going to draw that circle and ellipse for n o and any for 2 omega in the same picture it is as I will draw it in blue okay understood let us go through this quickly once again have you understood the circle and ellipse business for n o the polar plot is a circle because the refractive index does not depend on theta okay the polar plot means for this theta the tip of this pen will be at the value of n so if n is large it will be longer if n is small it will be like this okay so since n is n o is the same for all values of theta the tip of this pen is going to define a circle however for n e it is not going to define a circle with increase in theta it can either become smaller and smaller but when it becomes 90 degrees when it goes beyond 90 degrees it will start becoming larger again until it gets the same value here okay or it becomes larger and larger until it reaches 90 degrees and then it becomes smaller again until at 180 degrees it is equal okay that is how we get these 2 pictures the first one where n 0 is greater than equal to n 1 why am I saying n 1 and why am I saying n 0 sorry where n o refractive index for the ordinary ray is greater than equal to that for the extraordinary ray those crystals are called negative crystals and the second case where the refractive index for the ordinary ray is less than or equal to that for the extraordinary ray we call them positive crystals so far so good we have understood this diagram next we said now for the negative crystal I might as well have drawn it for the positive crystal it does not matter I am just showing you the example for the negative crystal now I want to draw the circle and ellipse for n o and n e not for the fundamental omega but for the second harmonic 2 omega okay so first thing we agreed upon is that this basic shape should not change a negative crystal will remain a negative crystal but then refractive index for 2 omega for the ore it is easier to understand the ore so it will extend to e ray as well will be smaller than that for omega so if I draw the surfaces for n o surface for n o for 2 omega it is going to be a circle with a smaller radius and what about the surface for n e for the second harmonic it is going to be a ellipse but a smaller ellipse so this is what I have drawn if possible try to not see the black picture try to see the blue picture only is the same as the black picture just smaller okay now comes the climax okay so far now see look very carefully there are 4 points at which n o for 2 omega and n e for omega have the same value 4 points at which n o for 2 omega the blue circle has cut the surface for any for omega the black dashed ellipse is that right I will show you one look at this point look at this point and this is where it is important to understand polar clause what it means the distance from the center that is the value and angular displacement from z that is theta okay so where the point where the blue circle and the black ellipse have overlapped is the point where their values are the same or in other words refractive index for the ore of 2 omega is equal to refractive index of e ray of the fundamental okay that is your phase matching condition remember what was phase matching condition that the fundamental and the second harmonic have to have the same refractive index here it is achieved okay we set out with the problem that you take a regular crystal the refractive index for 2 omega will always be less than refractive index for omega but then that problem has been circumvented because of birefringence due to birefringence you produce 2 rays the ordinary ray and the extraordinary ray alright while the refractive index of the ordinary ray does not change for change in theta refractive index for extraordinary ray does otherwise it would have been just 2 circles that would not overlap so because of this because you have 2 shapes you have an overlap some points overlap 4 points overlap to be more precise 4 values of theta at which the refractive index for the ordinary ray with frequency 2 omega has the same value as the extraordinary ray for light with frequency omega so phase matching condition is achieved alright experimentally how will I do it what am I doing here what I have discussed is this is z optic axis and this is the angle of incidence we are talking about increasing theta this way no need I can I can move the optic axis how will I move the optic axis I will rotate the crystal for the same angle of incidence let us say this is the optic well we are using this for the optic axis this is the incident beam and this is the optic axis of the crystal I rotate I rotate until the appropriate theta value is achieved where you have NO 2 omega equal to any omega you are done okay this is why we get second harmonic generation conveniently by simply rotating the crystal what we are doing essentially is we are changing the angle between the optic axis and the incident ray since Muhammad could not go to the mountain has come to Muhammad cannot change incident ray without messing up everything so we have just rotated the crystal now remember we are said something else we are said that the polarization of the second harmonic light is perpendicular to that of the incident light do you see why see phase matching condition has been achieved between what and what omega 2 omega I understand but ordinary ray of omega and sorry extraordinary ray of omega and ordinary ray of 2 omega and as we have said already ore and e-ray have perpendicular polarizations this is why second harmonic light has perpendicular polarization with respect to the fundamental polarization of second harmonic light is perpendicular to that of the fundamental okay so far so good everything is looking nice concluding comment for today is this nowadays at least for commercial setups you only have you all everybody uses crystals with a single optic axis uniaxial crystals there are crystals which are multi-axial fortunately people do not want to use them anymore I have used and believe me it is a pain multi-axial means you have to turn not only along this but also along this and along this sometimes so if that happens you have more parameters and the problem is you never know you turn in one direction you do not know whether you have reached the correct direction or not because the other two directions of the other direction that is there may be way off okay if you can get everything correct then you will get very good second harmonic generation but generally nowadays in commercial systems people do not like to use anything other than uniaxial crystals for the simple reason that there is only one control you turn in one direction change this value of theta you can imagine if there is another axis what will happen there will be theta 1 and there will be theta 2 and this theta is different from azimuthal angle that in principle it can go from 0 to 360 as I have shown 4 points right 4 angles at which you can get second harmonic generation do you observe that experimentally yeah you do not right usually you get only at 1 sometimes you get at 2 why because what it means is if you want to go to the other side incident angle instead of going this way the light has to go this way so it is equivalent to turning the crystal completely who will do that nobody does it. So generally there is one unique point where you can get second harmonic generation so that is the end of this 3 module discussion what we hope to have achieved in this is that we have got ourselves introduced to this fascinating world of non-linear optics a little bit without deriving anything and secondly we have at least understood one operation that we do regularly in lab and that is happily turn the crystal and see that you get nice blue light coming out. Now what we will do next is second harmonic is done in the next module we will try to discuss a little bit about some frequency generation and we will talk about 2 kinds of phase matching and then we will go on to discuss the next step of second harmonic generation of some frequency generation that is optical parametric amplification or optical parametric generation to start with see we are talking about generating some frequencies so far it is also possible to generate different frequencies difference frequencies that is where we will get to and then when we are done with that discussion we will try to discuss what is there inside the optical parametric amplifier and with that I think our discussion of instruments will be more or less done then we start discussing actual experiments we have already introduced pump probe but then what we can do is we can start with the classic experiments of amensuel then go on to other pump probe experiments then later on we are going to talk about experiments that involve this non-linear properties as well you want to talk about Raman experiments and all so we will need this later on as well.