 We are in the middle of our discussion of some frequency generation and second harmonic generation. We have studied the basics, leaving out most of the mathematics and most of the physics. So we are trying to develop a user's perspective of this rather complicated subject matter. Now today what we want to discuss is what are the issues that one needs to keep in mind when one wants to do some frequency generation or second harmonic generation. Not with regular continuous wave laser light but with ultrafast pulses. After all the entire course is about ultrafast phenomena and we need to be aware that it is important that we have to take care of certain parameters. If you want to do non-linear spectroscopy or even any non-linear optical process using ultrafast pulses. It is not as if you can take out a non-linear crystal from anywhere say your millennia is broken the continuous wave direct from solid state laser maybe you have a millennia that has gone bad. You take out that some frequency well non-linear optical crystal from there and you think that you are going to use femtosecond pulse using the same crystal it will not work. So today we are going to learn why it will not work and what is it that we need to keep in mind if you are going to do second harmonic generation some frequency generation so on and so forth using ultra short pulses. But before that let us do a quick recap of what we have learnt already. We have learnt that this is the expression for the intensity of second harmonic light. What are the factors there? Of course the second order non-linear susceptibility is 1 that should be as high as possible and as we have discussed without proving anything that there should be no centrosymmetry if this is to be non-zero. If centrosymmetry is there then in fact this is 0 forget about having a high value. Secondly we said that it is better to have a large value of I0 provided I0 is not so large that there is material damage in the crystal. And finally we worked with this term sin delta kl by 2 divided by delta kl by 2 whole square where l is the length of the crystal and delta k is the difference between the k vectors which boils down to difference in momenta of the combining light. And of course since it is sin square theta by theta square it is going to have a maximum at theta equal to 0. So we have come back to this expression shortly but what we said is that first of all l is limited by coherence length. We achieved delta k equal to 0 by angle tuning we are going to we have discussed it and we will do a quick recap in the next slide and we talked some about the polarization of the second harmonic being perpendicular to that of the fundamental. And again we are going to do a little more detailed discussion about that. And this is what we had discussed in the previous module in very great detail but once again not doing any real math. So what we learned there is that there are two things that are important here. It is not only that the non-linear optical crystal should have a high value of second order non-linear susceptibility but it should also be birefringent. What is the meaning of birefringence? The meaning of birefringence is that when light enters polarized light it should split into an ordinary ray or ore with the same polarization and an extraordinary ray or e-ray with perpendicular polarization. And the advantage of that is that you have crystals in which this refractive index for the ordinary and extraordinary rays are going to be different right and they are going to be different not only for the fundamental but also for the second harmonic. So we said that there are two kinds of crystals negative crystals for which N0 NO the refractive index for the ordinary ray is greater than or equal to that of the extraordinary ray and positive crystals in which refractive index of the ordinary ray is less than equal to that of the extraordinary ray. And the other thing that one needs to keep in mind is that for ordinary ray the refractive index is not dependent on the incident angle whereas for the extraordinary ray the refractive index is dependent on the incident angle. The polar plot for the refractive index of ordinary ray is going to be a circle and that for extraordinary ray is going to be an ellipse. So for negative crystals this is what the picture is going to be the one drawn in black. The circle solid circle is the polar plot for ordinary ray the dashed ellipse is the polar plot for the extraordinary ray. The extraordinary ray polar plot is contained completely within the polar plot for the refractive index of ordinary ray. In positive crystals it is the other way round because here NO is less than equal to NE whereas here NO is greater than equal to NE alright and then we said that let us consider a situation where the refractive index is less for the second harmonic compared to the fundamental then you get a similar set of circle and ellipse for the second harmonic but they are going to be smaller in size. And then since we have one ellipse we have an ellipse and we have a circle we see that in this particular angle of incidence the circle for the second harmonic overlaps with the ellipse for the fundamental which means the refractive index of the extraordinary ray of the second harmonic is equal to the refractive index of the ordinary ray for the fundamental for that particular angle of incidence and that is when the refractive indices are matched so phase matching is achieved the conservation of momentum is assured and that is when you get second harmonic generation okay. So this is what we had discussed in the last module I thought we will just recap once again because it is not very easy at least for us chemistry students to understand maybe so is there any question so far are you yes yes so I think what he is saying is that if the refractive index is very very small compared for the second harmonic compared to the ordinary ray then maybe there will not be any overlap if that is the case there will not be any second harmonic generation we will not get second harmonic generation in that case there has to be an overlap otherwise second harmonic generation is not going to happen it is not enough to take a non-centrosymmetric material just because your material is non-centrosymmetric does not necessarily mean that it is going to be NLO active okay. So in fact very few examples of materials are there where you are going to get second harmonic that is why they are so expensive and not so easy to make as well okay alright and then we said that it is possible so what are we doing what happens when we change the angle of the crystal essentially it is difficult to change the angle of incidence because then the entire optical path will change but if we rotate the crystal then we essentially play around with the optic axis of the crystal and thereby change the angle of incidence without having to change the direction of the light beam okay that is the trigger so angle tuning that is how it comes and we said that it is possible to have crystals that are not uniaxial but generally nowadays unless you have some very specialized application there are plenty of good uniaxial crystals positive or negative crystals which you could use so generally one would use uniaxial crystal unless there is some compelling reason to not do so and now I want to show you this table that you can actually find in Wikipedia there are other ways of representing the same thing but I like this table particularly because it introduces the terms that we are going to use in the next module when we talk about optical parametric generation and amplification it introduces the terms pump signal and idler and well why pump, why signal, why idler the reason is historic they are used like that it is like SPDF orbitals why are they called SPDF and not ABCD what is this, what is P remember we are digressing a little bit sharp principle diffuse something like that right so those are ancient terms which people have completely forgotten the relevance of here also more for historic reasons what we say is that this is basically a three color process right we are talking about some frequency generation special case of which is second harmonic generation right so we can say omega 1 omega 2 omega 3 right so what we are saying is the out of these three kinds of light the one with the smallest wavelength is called pump the smallest wavelength is called lambda p p for pump the one with the largest wavelength longest wavelength is called the idler okay it is nothing to do with laziness or activities of the light it is a name what it means is idler means out of the three beams that are there idler is the one with the longest wavelength that is smallest in smallest energy and the intermediate one is called lambda s s for signal okay so please remember this highest energy is pump intermediate energy signal lowest energy idler okay and also please do not miss the less than equal to science sometimes it is possible that say idler and signal might have the same values in whatever we have discussed so far it is not very logical to say that pump and signal will have the same value but idler and signal having same value we have encountered that already is not it when does idler and signal have the same wavelength in case of second harmonic generation lambda i equal to lambda s right so that is all that it is so please remember this lambda p less than equal to lambda s less than equal to lambda i that is all now the crystals are classified into type 1 type 2 sometimes they are classified as type 3 type 4 type 5 type 6 and so on and so forth depending on the relative polarizations of signal idler and pump in our case please do not forget so in the discussion so far pump is what is being produced okay right what is the meaning of pump in the discussion that we have had so far pump means some frequency or second harmonic right that is being produced so what this means is that if you have an a signal with well 0 degrees polarization idler with 0 degree polarization they are going to give you the some frequency which will have polarization at 90 degrees okay 0 and 90 are written as O and E respectively okay so as far as some frequency of second harmonic generation is concerned if you have same polarization of the combining light the 2 combining lights degenerate on degenerate then we have already discussed the case where you are going to have perpendicular polarization of the light coming out the some frequency of the second harmonic so that is called O O E kind of phase matching please do not get confused here the problem is since I have written pump first you have to read from right hand to side to left hand side opposite so type 1 phase matching is also called O O E phase matching meaning 2 light beams of 0 degree polarization are going to combine to give you some frequency of 90 degree polarization of course you might as well call it E E O means the same in fact you see O O E is called type 1 E E O is also called type 1 only if you want to differentiate between the polarization of the incoming light whether you want to specify that it is horizontal or it is vertical then you call this type 1 and you call this type 8 okay and the discussion that we have had so far is actually of type 1 phase matching right and will you agree with me if I say that when we talk about second harmonic generation then it is going to be type 1 I will clarify the statement little more maybe expand the statement little more shortly but in the discussion that we have had so far will you agree with me second harmonic generation unless you separate the beams rotate the polarization of 1 for some reason the light that goes on goes in it has the same polarization right you are mixing light with itself the input light with itself so that is why if it is O then O O E is what you will get if it is E then E E O is what you will get but there is another kind of polarization type 4 and type 5 type 5 is also called type 0 where you have E E E O O O that means there is no change in polarization when will that happen when will no change in polarization happen when a circle overlaps with a circle rectangle overlaps with sorry not rectangle ellipse overlaps with ellipse right right or wrong why did we get O O E earlier because the refractive index polar plot for refractive index of the O ray of the fundamental sorry E ray of the fundamental overlap with the polar plot of the O ray of the second harmonic okay now relative sizes can be different right depending on which material it is depending on that it is possible that 1 rectangle 1 ellipse will cut another ellipse okay then you get E E O kind of thing and then if you mix 2 kind 2 different lights now if it is 2 different rays it can be omega 1 plus omega 1 but suppose you have taken that light and you have rotated the polarization by 90 degrees and combining then you are going to have O E then if it is type 3 or 2B you will get E polarization if it is type 6 or 2B or 3B you will get O polarization okay so there can be many combinations depending on what kind of whether the crystal is positive or negative and what kind of crystal it is there is a compendium of which crystal is of which type in this book that we are studying you can look it up okay but it is important to know this because polarization does have an important role to play in any non-linear optical application that you want to work with okay right so these O E or E O E this you get for negative crystals and this are usually obtained for positive crystals this way you can look at many different combinations. Now let me show you some example so these are some examples KD star P what is that what is KDP what is KTP homework I will not tell you you have to find out the only thing I will tell you that star means it is deuterium instead of hydrogen but please find out what is KDP what is KTP what is LBO BBO perhaps you know beta-varium borate but you should know what this means and what is this LI NBO 3 lithium niobate right lithium niobate so these are all different kinds of facemaking crystals last 3 are type 1 first 2 are type 2 okay these are examples there are other examples as well now the question is which of these will you use will you use type 1 or type 2 as you see what some other parameters are there that become more important than type 1 or type 2 when we want to talk about using ultrafast pulses okay so to understand that once again let us go back to this expression that we had written a little earlier which parts of these this expression are going to be helped by ultra short pulses and which parts are going to be made more difficult first of all we all have we have discussed earlier that ultra short pulses have high intensity that is one feature and second feature is that it has a spectral bandwidth spectral bandwidth is that good or is that bad high intensity is good isn't it because here you have this factor of i0 so laser pulse if you use ultra short pulses then at least this i0 square is going to be high anyway and we know why we have discussed that already now spectral bandwidth is it good or is it bad we are telling you it is bad why is it bad what is the discussion we have had so far we said remember the k vectors yeah this k is written here so the entire discussion was based on the fact that you have to have an exact phase matching that is delta k has to be equal to 0 now what do you have in a pulse pulse light is it monochromatic it is not what does that mean it has many colors many different what we call plane waves a plane wave is essentially a monochromatic waves where you have these parallel planes that define different amplitudes different phases okay so this is not a plane wave we will discuss maybe in the next module what it is but this is a problem how are we going to achieve delta k equal to 0 when there are so many ks you get the problem it is a problem okay so the problem is that you have to somehow achieve phase matching for the entire band how will you do it well nature gives us a way out fortunately and from this expression can you tell me what the way out is what is a natural way out natural that word is a hint if I want to plot this function i2 omega what will I get will I get a delta function then what is the factor that yes what is the important factor in this discussion sin square theta by theta square so that has some finite width is not it remember in our discussion of line width and all sin square theta by theta square is the minimum line width that has to be there even if you have two specific states and that is what gives you natural line width to spectral lines so this is the plot really remember so the thing is this here at this point sin square theta by theta square equal to 0 but you do have a width all right so even this is not exactly 0 so if you consider the width of this function and all it turns out that it is sufficient you can manage to get phase matching provided delta k into L is less than equal to 2.78 okay no this 2.78 just comes from you know numerical solution of this problem I cannot write a formula and say that this is why it is 2.78 difficult to do that right but if you look at the pulses it turns out that your okay provided delta k into L is less than equal to 2.78 okay now what is the meaning of delta k once again delta k means basically difference between the k vectors right so if you have many k vectors you will have several delta k values within this limit so here if you read this ancient paper you will see that okay what does each k characterize it characterizes one particular plane wave right now that particular plane wave is also characterized by a characteristic frequency so what this paper does from 1968 is that it starts from this delta k dot L less than equal to 2.78 and works out the expression for delta nu nu is something that we understand better and nu is something that we work with a little better right so we will not do the derivation I will just show you the expression delta nu where delta nu is nu minus nu 0 what is nu 0 the central frequency so delta nu minus nu 0 is only half of the band right so nu minus nu 0 has to be less than equal to 0.221 by L multiplied by lambda f d and d lambda f f stands for fundamental minus lambda second harmonic d and d lambda second harmonic reciprocal of that what is d and d lambda what is n here refractive index right and we know that refractive index is a function of frequency d and d lambda is the rate of change of wavelength with respect to sorry rate of change of refractive index with respect to wavelength that is all okay gradient okay so this is what it is so now from here if you know which fundamental wavelength you are dealing with and d and d lambda for fundamental as well as second harmonic can be known experimentally from for the material and I hope you understand that d and d lambda for fundamental or d and d lambda of second harmonic is going to depend solely on the material that you are using okay and it is not very difficult to work are you take thin slices of the material see how much of change in refractive index is there and you can work it out so from there for a 1 centimeter kdp crystal it turns out that delta nu is 20 centimeter inverse whereas for 1 centimeter lithium niowate crystal delta nu is only 0.5 centimeter inverse but then I cannot just stop here I have to say which wavelength it is it will depend on lambda fundamental lambda second harmonic also so the wavelength that we are working with is 1064 nanometer what is the claim to fame of 1064 nanometer 1064 nanometer is a magic number what is it yes is a fundamental emission of new NDAG right one of the most popularly widely used lasers 1064 nanometer you will recognize the second harmonic better 532 nanometer so if I want to convert 1064 nanometer to 532 nanometer which material would be better if I am using a pulse should it be kdp should it be lithium niowate let us close this module on this note we will come back in the next module and we will discuss from this this point on