 So, now today onwards we start discussing laser basics and laser as we know is light amplification by stimulated emission of radiation and stimulated emission of radiation is what we want to start with. So, we start talking about something that is very fundamental we have discussed it in our spectroscopic course as well and I am sure most of you would have studied it in your MSC curriculum at least but still we will revise it in case first of all we need to remind ourselves and secondly in case we had any lacuna in understanding that hopefully we will get sorted out now. So stimulated emission is what we want to start talking about and the reason why we want to talk about lasers is that it is central to any study of ultrafast dynamics it is central that you use a pulsed laser as light source so before we go to the pulse part of it we should at least know how a laser works then only when we talk about how pulsing is done and all it will start making sense. So what we are doing now is actually like the part before time 0 in our transient absorption or up conversion measurement we are going back to the very basics and in the next couple of modules or 3 modules will be about that and for this actually one can read from something as fundamental as Macquarie and Simon's book physical chemistry molecular approach by Macquarie and Simon. So this is a standard book for undergraduate and MSC classes as well the discussion here should be enough for the next couple of modules and then we move on to more sophisticated books right. So what is stimulated emission those of you who have studied the interaction of radiation of matter from a quantum chemical perspective at least a semi classical treatment of it at least might remember that the way it was done is that you consider 2 energy levels let us call them 1 and 2 1 is the lower 1 2 is the higher 1 and we said we ask the question what happens when a photon of appropriate frequency is incident on this system. So actually 2 things can happen what is very clear to us is this photon might come and cause an upward transition this I think everybody understands is an absorption process. Now what is the condition for the photon to be absorbed it must satisfy the Bohr residence condition h nu 1 2 let us say this frequency is I am writing 1 2 specifically because I want to highlight the fact that the frequency of the photon is such that it is energy matches exactly the energy gap between the 2 states involved. So when nu 1 2 is incident photon of nu 1 2 is incident on the system we understand very clearly that absorption is going to take place. What we might not understand to start with is that the opposite phenomenon might also be brought about by a photon this is an emission but it is not an emission by itself this is called a stimulated emission this is absorption why is it that a photon would cause an emission of another photon because if you go back to the formulation of the problem in a semi classical limits the light here actually should not even say photons if you are talking in semi classical terms because there the molecule is modeled using quantum mechanics light is modeled as a wave that is why it is called a semi classical treatment in the first place. So let us think like this light which is a wave acts as a perturbation here you have 2 energy levels you have some population distribution between them what light does is that it disturbs the level and causes a mixing. So when I say mixing n 1 has to be mixed with n 2 n 2 can also be mixed with n 1 mixing of states in quantum mechanics is equivalent to a transition in spectroscopy. So here when light comes and causes the transition it can do it both ways to put it very qualitatively this is stimulated emission and when stimulated emission takes place the light that comes out you can write like 2 curly arrows first of all it is important to understand that light that came in is actually conserved and some more light comes out now I have no option but to go back to photons if one photon causes the transition then 2 photons come out if there is stimulated emission and if there is no other loss in some other way okay. So this gives you a multiplication of number of photons this is very important to understand and not only that the light that comes out is correlated with the light that caused the downward transition that is why there are several properties that are associated with light that comes out as a result of stimulated emission may not always be there for spontaneous emission but we have not even talked about spontaneous emission yet. So let us see what kind of correlations there will be between light that comes out and light that goes in in a stimulated emission process first of all the frequency of the light that comes out is exactly the same nu12 right. So you get monochromaticity secondly then this is a curious property let us say this is your sample here light comes in from this direction original light goes in goes out in this direction the light that is that comes out as a result of stimulated emission will follow the same path. So in stimulated emission you get directionality unlike light that comes out from regular light sources for example right generally the light sources that we have or fluorescent molecules and all that we have they are going to emit in all directions if it is spontaneous you know there is nothing to drive the direction here the emission is actually driven by the light that comes in the light that produces the perturbation we are not doing the math here but there is something called transition dipole moment that ensures that not only is it directional but also something else is there polarization of the light is maintained if you put in vertically polarized light and if there is no rotation and all then stimulated emission light that comes out is also vertically polarized okay and another important property is coherence coherence means not only are the two light waves monochromatic monochromatic means what exactly same wavelength polarized that means the oscillations are in the same plane that is not enough what happens is that they are in step light that comes out as stimulated emission is exactly in phase with the light that causes it and these are the properties that make stimulated emission a suitable candidate for light amplification and obtaining lasers with the properties that we know that most lasers have okay and this is the simple formulation but there is something else that we need to discuss here which might sound a little off topic but actually it is not as you will see by the time we are done Einstein did a kinetic treatment of this process because something that is obvious here is that we are not discussing the whole thing if you only talk about induced photo induced processes induced absorption if you want to emphasize the induced part of it and induced emission or stimulated emission then one thing that you are definitely leaving out is spontaneous emission but Einstein said that that is not practical because spontaneous emission does take place we see it all the time all around us we cannot say that only stimulated emission takes place even though in the realm of semi classical treatment using time dependent perturbation theory you only consider induced processes induced absorption induced emission. So what Einstein did was that he brought in this third process which might actually be more obvious to us than stimulated emission he brought in spontaneous emission okay. So the way I have drawn it here in the way that I have drawn it spontaneous emission and stimulated emission actually have the same wavelength because I have only 2 energy levels and nothing else but properties that will not be seen in stimulated in spontaneous emission what will be seen in stimulated emission are those that we have written here 234 okay. Of course this is a discussion that was performed several decades ago nowadays people are working on how to get directional spontaneous emission. So if you did work by Lakovic they have made some progress in it there is something called I just told you that for amplification stimulated emission is a good candidate but actually we are going to discuss later when you try to make a laser and try to get amplification of spontaneous emission one big problem that shows up is AC and again that term AC might be a little confusing because AC is there in laser as well you just take out the L and take out the R what you are left with is AC but when we say AC then generally we mean amplification of spontaneous emission you have to kill amplification of spontaneous emission if you are going to get a good laser. If you work with homemade lasers and if you try to actually get the lasing done there AC can turn out to be a threat and the reason why it can turn out to be a threat will come to shortly okay but let us do Einstein's formulation here. What Einstein did was a very simplistic kinetic formulation all of us have studied chemical kinetics I hope at least in 11 12 so there what we know is we know how to write differential equations what is the rate what is rate it is something like dx dt okay so we will write the rate equation and to start with we will not even write the dx dt part we will only write with W the right hand side without W we will only write the RHS okay so let us say we consider the first process absorption see absorption is sort of like a an elementary bimolecular reaction right you can think it is a reaction between a photon and molecule okay so if it is bimolecular then its rate will depend on the product of concentration of the molecule in state 1 and concentration of light okay so what will the rate of upward transition be it will be n1 where what is n1 n1 is the population what rather say number number of molecules in state 1 multiplied by after multiplied by concentration of photons that is usually given by something called energy density I will write it like this rho 1 to nu it is a little bit of an overkill here because the moment I say nu it appears that 1 2 is taken care of actually it is not energy density rho everybody knows this density just to emphasize that it is density of energy and not of matter we write nu in bracket and 1 2 subscript is there to denote what is the energy gap okay is any other thing here for the rate concentrations have both taken care of concentration of light concentration of molecule the only thing that is left is the rate constant and Einstein for whatever reason wrote this rate constant as b I can write 1 2 alright so b 1 2 is the rate constant for absorption it is called Einstein's b coefficient but we can come to that later rho 1 to nu is the energy density can anybody does anybody remember where we encountered energy density in some very highly celebrated blackbody radiation right in blackbody radiation we had studied energy density and there is some expression for it right from Planck theory so I will write that expression here so that we can use it later on that expression is and also you might want to write it down because this is going to be useful I can write like this rho 1 2 of nu is equal to 8 pi h by C cube nu 1 2 cube e to the power divided by e to the power h nu by k t this k is not rate constant this case Boltzmann constant minus 1 this expression for energy density is going to come extremely handy in the next 5 10 minutes alright so this is energy density that is known number of molecules in state 1 is n 1 and the rate constant is b 1 2 so this is the rate of formation of n 2 from n 1 rate of absorption what about this process again and now I can start writing from the beginning because we know basically what it is again it is sort of a bimolecular reaction between energy and matter so there will be some rate constant and again I will write b and this time instead of 1 2 I will write 2 1 of course I will have rho 1 2 nu there is no point in writing rho 2 1 nu because energy density of a particular frequency will be the same right that is the property of light actually and then multiplied by n 2 or n 1 n 2 what is n 2 n 2 is the number of molecules in state 2 okay now what about this spontaneous emission process that is like an elementary unimolecular reaction because no light is there molecule has been excited fine but it has been excited by the absorption process when it emits it emits on its own so here the only quantity that will be important is n 2 and the rate constant here is written as a 1 2 Einstein's coefficient for spontaneous emission so here we have 3 Einstein's coefficient a 1 2 is Einstein's coefficient of spontaneous emission a 2 1 sorry as you will see in a while it will not matter a 2 1 is Einstein's coefficient of spontaneous emission b 2 1 is Einstein's coefficient of stimulated emission b 1 2 is Einstein's coefficient of you can say stimulated absorption okay so this is the formulation of Einstein problem now we will go ahead and actually write the rate equation but again we are going to use something that we have studied in chemical kinetics and that is steady state approximation remember steady state approximation when you have something like a reactant going to a product p through an intermediate i what is steady state approximation d i d t equal to 0 i in third bracket of course concentration of the intermediate is 0 right now what will happen how do you get that concentration equal to 0 because what you say essentially is that the rate of formation is equal to the rate of well deformation would sound strange rate of breakdown so here we can say that at steady state rate of formation of 2 is equal to rate of depopulation of 2 which means b 1 2 rho 1 2 of nu into n 1 should be equal to a sum of b 2 1 rho 1 2 nu into n 2 multiplied by sorry plus inside brackets a 2 1 n 2 okay let me write it what I am saying is at steady state d n 2 d t equal to 0 okay now let me write the expression for d n 2 d t b 1 2 rho 1 2 of nu n 1 then I should write if I am going to follow from here I should write minus b 2 1 rho 1 2 of nu n 2 minus a 2 1 n 2 is equal to 0 okay I will step jump a little bit and write something like this b 1 2 rho 2 rho 1 2 nu into n 1 is equal to b 2 1 rho 1 2 of nu multiplied by n 2 plus a 2 1 into n 2 clear now we go ahead and try to find a solution for this one well try to find a solution means what we will do is we already know the expression for rho 1 2 right we will try to find the expression for rho 1 2 and from there we will try to get an idea of what a is and what b is to the maximum extent possible so what I want to do is I want to collect the terms in rho 1 2 so whatever is there in rho 1 2 we can bring to left hand side. Let us proceed rho 1 2 of nu multiplied by b 1 2 n 1 minus b 2 1 n 2 is equal to a 2 1 n 2 simple so write rho 1 2 nu is equal to a 2 1 n 2 by b 1 2 n 1 minus b 2 1 n 2 is this correct now is a good time to write the other expression once again and start comparing what was Planck's expression rho 1 2 at nu equal to 8 pi h nu 1 2 by c whole cube divided by e to the power h 1 2 by k t minus 1 right so what we will try to do is I will try to write this expression in black in such a way that it will look more or less like the expression in blue and then we will just compare the terms how do I do that second term has to be 1 minus is already there so how do I get second term equal to 1 if I just divide numerator and denominator by the second term then I will get 1 so that is a good seems to be a good starting point so I can write a 2 1 well n 2 by n 2 is equal to 1 so I will not write that divided by b 2 1 that is what I have in the numerator are you following what I am doing I am dividing numerator as well as denominator by this and in the denominator I know the second term is very easy that is minus 1 what is the first term first term is b 1 2 by b 2 1 multiplied by n 1 by n 2 right now see we have 2 energy levels 1 is lower and energy lower energy 2 is higher in energy so we actually know what this n 1 by n 2 is do not we Boltzmann distribution what is n 1 by n 2 remember n 1 is lower energy n 2 n 1 is the population of lower energy level n 2 is the population of higher energy level so it will be e to the power h nu 1 by h nu 1 2 divided by k t right we are more used to writing the higher energy one in the numerator that is why you get e to the power minus h nu by k t here we have written the lower energy population at the top so and then now see it is folding unfolding in front of your eyes this n 1 by n 2 turns out to be e to the power h nu 1 2 divided by k t okay now compare this expression with this expression what is the first thing we get first thing we get is that this b 1 2 by b 2 1 has to be equal to 0 right so as to get the right denominator so first thing we learn is 1 sorry sorry b 1 2 by b 2 1 if it is 0 then we can go home is equal to 1 or I can write like this b 1 2 equal to b 2 1 I do not need 1 or 2 anymore I can just write it as b so this is your Einstein's b coefficient and the messages carries after such a simple exercise is profound what you are saying is that for an induced process for a given two two level system rate constants for the upward and downward processes are actually the same okay and this is something we can remind you of what we have discussed earlier remember we had said sometime that your quantum yield fluorescence quantum yield is related to epsilon right epsilon is something that you measure from absorption spectrum right so and that is an intrinsic quantity it has got to do with probability of transition from lower state to higher state emission quantum yield talks about probability of transition from higher state to lower state I hope you have not forgotten that we had said these two are correlated if epsilon is high for a molecule then quantum yield is also expected to be high there is a reason why I say expected to be and not is we will come to that shortly but do you understand that they are expected to be high why because of this because b12 and b21 rate constants they essentially are measures of probability of transition probability of transition from 1 to 2 and from 2 to 1 so what we are saying is for induced processes the probabilities are the same no matter whether it is an upward transition or downward transition that is why your emission quantum yield and epsilon should be correlated I am not saying equal correlated proportion has everybody understood is everybody comfortable with this I am not comfortable because if you remember all the emission that we have talked about earlier is spontaneous emission and what we have shown here is that b12 equal to b21 here the process of emission that we are talking about is stimulated emission so if I am going to say what I just said then I better establish some kind of a linear relationship between stimulated emission and spontaneous emission as well of course for that I do not have to go very far it is right there in front of our eyes look at the numerator here look at the numerator here what you see from here is now there is no need to write 1 and 2 so I will just write a okay so I see that a by b is equal to 8 pi h nu 12 cube by C whole cube so a is actually proportional to b all right so it is okay if I am talking about spontaneous emission as well because after all spontaneous emission well rate constant of spontaneous emission Einstein's a coefficient varies linearly with Einstein's b coefficient so there is no problem well there is more that we want to say about this a and b business but let us do that in the next module.