 We are back and do I have an answer which is better while working with utter short pulses is KDP better or is lithium niobate better KDP is better because see the problem is this what it means is that if delta nu is more than 20 centimeter inverse you will not get second harmonic generation for KDP but if delta nu is more than 0.5 centimeter inverse you will not get second harmonic generation for lithium niobate and we know very well that the characteristic of an ultra short pulse is a large bandwidth shorter the pulse bigger the bandwidth right. So if you are going to use a short pulse femtosecond pulse and use lithium niobate to generate second harmonic wavelength what will happen your modal wavelength is 1064 nanometer for fundamental so you are going to generate 532 nanometer fine but the bandwidth will be very small yes if bandwidth is very small then we have studied transform limited pulses if bandwidth is small then what does that mean what does that how does that affect the pulse width pulse width become becomes large right remember the delta nu multiplied by delta t that would be a constant depending on what shape the pulse is and the physical reason for that is if the spectral bandwidth is large that means a large number of longitudinal modes have been locked to produce the pulse greater the number of modes locked shorter is the pulse wider is the spectrum. So now after second harmonic generation if the spectrum has become narrow just because your delta nu is small for the crystal then automatically what will mean is that in the second harmonic light very small number of modes are locked right when width is small that means fewer number of modes would be locked compared to fundamental which would mean that the pulse width would be significantly larger compared to the fundamental right even 20 centimeter inverse is not very large but at least it is much better than 0.5 centimeter inverse so you want a crystal so that is the first point we are we want to make here you want a crystal for which delta nu is going to be large as large as possible so that your pulse does not become temporarily broadened specially it will become narrower but at the same time temporarily it will become broader if delta nu is small. So in order to ensure that your ultra short pulse remain ultra short even after frequency doubling you want delta nu to be as large as possible right and what determines delta nu the material. So this is an important point here you want to use a material for which delta nu is large okay while working with ultra short pulses okay. So it is a very complicated thing first of all you want a large value of second order nonlinear susceptibility right chi second order then you want the crystal to be bidefringent then when you are working with ultra short laser you want the delta nu to be larger so we are getting into more and more restrictions and in fact there will be one more before we complete this discussion. So far we have talked about this bandwidth business next we want to discuss another aspect of ultra short pulses and that is group delay we will say what group delay is but the central theme of this entire discussion is something that we had discussed several modules earlier the issue is our ultra short laser is not a single mode laser lot of longitudinal modes are actually locked to produce the ultra short pulse okay. So whatever discussion we had earlier during mode locking that becomes the determinant in the discussion we are going to perform now. So when we do the experiment we do not even think we tweak this we tweak that and things happen but a lot of effort by many people has gone into the system before we could use it as a toy and it is important that we understand we do not need to know all the math but we should at least understand the principles otherwise if you have to design experiment if you do not know these we will end up taking out a crystal from anywhere and trying to do second harmonic with ultra short pulse and it will not work okay. So second factor group delay now see we have discussed earlier that plane waves are characterized by their phase velocity right VPH equal to omega by K is equal to C divided by refractive index at that value of omega we have talked about this earlier yes. So now how do I define pulse light pulse light you said is actually not used you did not say anything I said but you know that pulse light is a mixture of many plane waves right. So the amplitude wave function of pulse light can be written like this you might as well write a summation if we write an integration because it is easier to arrive at the next step if you do an integration you can do it numerically as well okay. So let us understand what we have written forget about forget about this integral for the moment we will come to that what is this a k a at k multiplied by e to the power i kz by minus omega t that is the expression of a plane wave we have written that right and our other short pulse comprises of many such plane waves so what we are doing is we are adding up all the fields so psi as you know is an amplitude and it depends on z as well as t what is z z is the direction of propagation of light okay if light goes in this direction then this is z direction of propagation of light and of course it is dependent on time as well so what we have is psi at some value of z and some value of t is an integral of this field for a plane wave over the entire range of plane waves so integral dk and lower limit is k0 minus delta k upper limit is k0 plus delta k so what is k0? k0 is the k value of the modal k0 is the let us just say k0 is the modal k value so delta k on this side delta k on that side we are working with a symmetric kind of distribution okay how do you do it we are not going to do the entire integration any integration enthusiast is welcome to do so I will just tell you how it starts and I will show you what the final answer is so to work out this integral what is done is omega so as I said earlier every plane wave is characterized by a k vector and an omega right a characteristic k vector characteristic omega so we can pretend as if omega is a characteristic of k so we can write omega for a particular k value that is written as omega 0 again the modal omega plus d omega dk0 multiplied by k minus k0 does this ring a bell have you encountered an expression like this somewhere in spectroscopy fundamental spectroscopy not in this course yeah where a little louder please very easy question as demonstrated many times I do not ask difficult questions yeah where have you encountered why Raman what about vibration when you derive the vibration selection rule how do you write the dipole moment mu equal to mu 0 plus del mu del x multiplied by x so that is what it is okay this is a rate of change of omega with respect to k that is multiplied by k minus k0 it works because k minus k0 is a small quantity okay this is how it is expanded and then after doing the integration now I will show you the final expression please do not get scared we will demonstrate that it is not very difficult expression so this is the final answer nothing to be scared of as you will see psi at z and t particular value of z particular value of t turns out to be 2 a at k0 m a at k0 means amplitude for the modal k value k0 neglect this thing for the moment the last factor is e to the power i k0 z minus omega 0 t that is quite simple isn't it what is e to the power i k0 z minus omega 0 t that is the same expression you have here instead of z you have written instead of k you have written k0 instead of omega you have written omega 0 so that is the expression for the once again the plane wave at the center of the distribution alright I am talking about this one I wish I had better control over my fingers but I do not alright this is what I am saying so do you see that that is what comes from the plane wave at the center of the distribution and this okay it is the amplitude it comes from the integration what are we left with what are we left with yeah what kind of a function is that earlier where sin square theta by theta is it that or is it something else is sin theta by theta so what is the difference between sin square theta by theta square by sin theta by theta more or less similar looking functions but there is a difference yes sin square theta by theta square is always positive but this goes negative value not too much negative but it does go negative so this is the shape okay now this whole thing here is the amplitude of the wave packet let the term wave packet not scare you some people write wave packet as one word some in some books most of the books wave packet is written as two words I prefer to write these things as one word always nanoparticle is one word for me wave packet is one word for me because I am influenced by a television series that I saw as a kid that was on body line bowling you know body line Douglas Jardin Don Bradman so there there was this scene where this British reporter is trying to send a message by telegram and he fights with the clerk at the telegraph office because he wrote body line which was a new word at that time there was no such word earlier so the clerk insisted that this is two words so every time you write body line you have to pay for two words and the journalist insisted that no it is a new word it is one word so somehow that got into my head and I like to combine words whenever you can so for me wave packet is one word you write whatever you want what is the meaning of wave packet what is the wave packet my favorite answer to this question sounds almost stupid a wave packet is a packet of waves it sounds very strange mundane and but actually that is what it means you combine a lot of waves right that gives you a packet of waves and you combine them in such a way that at one particular point they are all in phase then as you go out on either side they go out of phase and that is why you get a shape like this okay how do you generate a wave packet we are going to discuss this later suppose once again it is related with that spectral bandwidth of ultrafast pulses ultra short pulses suppose you excite something using an ultra short pulse excitation starts from v equal to 0 of the lowest of the ground electronic state let us say right what is the destination can you say that we only we only populates a v equal to 3 v dash equal to 3 of a s 1 you cannot because there is a distribution right some molecules will be excited to v dash equal to 3 some might be excited to v dash equal to 2 some will be excited to v dash equal to 4 and so on and so forth and then that will form a coherent wave packet and coherent wave packet dynamics is an important problem that has been studied for ages in ultrafast spectroscopy we are going to discuss it in this course as well but for now this is a wave packet okay ultra short pulse and this is a shape that you get of the amplitude amplitude of the wave packet now this entire wave packet it is not stationary it is not a standing wave the entire wave packet moves okay and it moves with what is called a group velocity given by d omega dk 0 do you get the significance of d omega dk 0 look at this function where will the function be equal to 0 z equal to z minus dw d omega dk 0 t is equal to 0 right that is where this is going to be equal to 0 is that right what is z by t so what we are saying essentially is that z equal to d omega dk 0 t okay so what is z by t I got t from here and I write t here z by t is equal to d omega dk 0 this why do you call that group velocity all of a sudden what is z what is z last letter in English alphabet yes but in this context what is z here to speak loudly it is a big room very few people are there I cannot I am a little short of hearing yeah what is z it is the direction of propagation of light so what is z by t distance covered in direction propagation of light per unit time that is velocity right so z by t essentially is the velocity with which this entire wave packet moves okay d omega dk 0 this is called a group velocity not group delay yet delay comes later when we talk about delay you have to talk about two kinds of light the two kinds of light in this context are fundamental and second harmonic we will come to that for now what we have been able to do is we have been able to define a group velocity of a wave packet so group velocity is d omega dk 0 okay and just for the record fundamental and second harmonic have different group velocities and that creates another problem one problem we have discussed already is delta nu the other problem is that if fundamental and second harmonic travel with different group velocities then this is a situation you get as time passes they move away from each other so the question is what is the difference in times taken by fundamental and second harmonic to travel the entire length of the crystal okay once again without derivation I will give you the expression this is what it is delta t is given by 1 by c dnf d lambda f multiplied by lambda f where f is fundamental equal to dnsh d lambda sh multiplied by lambda sh I might as well have like the previous slide I might as well have written d and d lambda then put a bracket and written f as subscript that would have far as been better okay so you do not want too much of delta t that is the issue that to travel together for the wave packet to sustain so now what do you want you want the delta t to be smaller than the laser pulse duration yes so in other words we can say that delta t should be almost equal to 0 time required so as long as they are within the crystal they should be together the fundamental and second harmonic so if delta t is to be equal to 0 work with this expression equate that to 0 what do you get yes while doing that do not forget something and you equate that to 0 is there some relationship between lambda f and lambda second harmonic and the better would be what is the relationship yes lambda f equal to 2 multiplied by lambda second harmonic so putting this expression into this what do you get you get something like this okay this is another condition d and d lambda of fundamental should multiplied by 2 should be equal to d and d lambda multiplied d and d lambda of second harmonic d and d lambda right rate of change of refractive index with respect to wavelength for second harmonic should be equal to twice d and d lambda for fundamental okay that is when your delta t is going to be 0 and delta t should be small in order to get good second harmonic generation using an ultra short pulse right so once again let me show you some typical values delta t for lithium niobate is 6 delta 3 for lithium irate is 0.7 delta t for KDP is 0.08 so which is the best out of this three from this point of view definitely KDP okay so the take home message from this module and the last one is that one needs to be careful about several things before one can think of doing second harmonic generation or some frequency generation of using ultra short pulses first of all you have to choose the right material and there are so many parameters right we have said this once already large value of second order non-linear susceptibility birefringence delta t has to be close to 0 and what else what is the point we discussed just before this delta nu delta nu has to be as large as possible okay one more thing that we have not really said explicitly do you want to use a crystal that is very large or do you want to use a crystal that is small I mean that is not too thick you want to use a large path length within the crystal do you want to use a not so large path length you want to use a thin crystal because here you are talking about delta t delta t is going to be longer for a thicker crystal see delta t here is given in picosecond per centimeter right so if you use two thicker crystal first of all they will be dispersion so your pulse will become broad and this delta t is also not going to help so these are several parameters that one needs to worry about when one wants to do non-linear optical manipulations using your ultra short pulses right so that brings us to the end of the discussion that we intended to do today but let me just give you a preview of what we are going to discuss the next day see so far we have been talking about some frequency generation right combining two photons of small energies to produce a photon of larger energy and that is easily understood is it possible to have a photon of large energy split into two photons of smaller energy in a non-linear optical material the answer is yes it is just that it is a much more difficult process compared to some frequency generation and that splitting into two kinds of light with smaller energy longer wavelength has a name that name is parametric generation or optical parametric generation okay so it is really very small like 1 in a million kind of probability and the phenomenon is not easy to understand so far even though we did not do the derivations we more or less understood what we are talking about because we can think of real life analogies and all difference frequency generation is actually not easy to understand because it deals with things like quantum entanglement in fact parametric generation is used to make what are called correlated photons or single photons or photon pairs not of any use to us as such but they have they are used in sophisticated optics experiments we are not going to go into that in this course but the point is it can happen what we want to learn is we are saying it is a 1 in a million kind of probability to get parametric generation is there any way to increase the intensity so now see what will happen so far we are dealing with a situation where signal and idler meet at the non-linear optical crystal and the pump is generated in fact when I say it like this it sounds little strange isn't it because when we say pump we think the pump should go in that is what is going to happen in parametric generation the pump goes in and splits into signal and idler so signal idler both have longer wavelength lesser energies so this is a good way of generating light of so using visible light it is a good way of generating say IR light now the question is how do we get a higher intensity two ways one is do this parametric generation inside a resonator so if you have parametric generator in say resonator means a laser cavity right if you do multiple reflections then it happens multiple number of times you are going to get some kind of increase in the output the other way is to do what is called optical parametric amplification that is what we use in our lab right and that is a two-photon process it is not only the pump that goes in into the non-linear optical crystal it is pump and signal and what you generate is signal and idler okay that is what we are going to discuss next day so I think next module is going to be very very sketchy principle of optical parametric generation and optical parametric amplification you might not talk too much about optical parametric oscillation and module after that will be a discussion of our topaz manual what is there inside the optical parametric amplifier that we have how does it work those of you who have worked with topaz can you tell me what are the different kinds of light that are there inside there are several crystals is not it and what do you do you do second harmonic generation in some crystals you do something else in some other crystals what is that don't you generate white light so that white light now by angle tuning and all one particular frequency from that white light well of course it is modal frequency that gets amplified that is how an optical parametric amplification works so that is what we are going to sort of learn in the next couple of days and we will perhaps say a few words about so far we are talking about collinear geometry in fact even our topaz has collinear geometry but there are certain advantages that come if you use non collinear geometry and in context of optical parametric amplification that instrument is called a NOPA non-linear optical parametric amplifier so if possible in the next couple of modules or in the next three modules this is what we are going to cover let us see whether we can give you some brief idea sketchy idea about NOPA as well okay today we stop here.