 So, now we want to continue our discussion on two level systems but what we would like to do is well we are continuing the discussion so there is no but anyway we will go back and write down the expression that we had arrived at at the end of the last module and that is a by b equal to 8 pi h nu12 divided by c whole cube what does nu12 stand for once again this frequency nu12 what does it denote? It denotes the energy gap so from this expression what can we say about the relative propensity I do not want to use the term probability right now relative propensity of spontaneous emission and your stimulated emission first thing that we have to recognize is that a is never equal to 0 if b is not 0 sounds very simple because a is some constant multiplied by b right and that constant is not 0 so if b is not 0 then I cannot be 0 and if b is 0 then what does it mean? That means even absorption will not take place right we just proved that b is the same no matter whether it is absorption or stimulated emission so that means if b is 0 the transition is forbidden okay in a spectroscopic course we have worked out the relationship between Einstein's b coefficient and the square of transition moment integral okay so that is equal to 0 so transition is not even there if transition is not there what are we talking about? So in all this discussion we talk about situations where at least b is not equal to 0 if b is equal to 0 then again we can go home okay so that is the first thing sounds very simple but actually this is what vindicate Einstein treatment it is important and this is something that is going to come handy in our discussion towards the end of this module a is never equal to 0 if b is not 0 right seemingly simple straight forward might seem obvious but extremely important profound. Second thing is this is a by b ratio when is a by b ratio more when is a by b ratio less on the right hand side well 8 is a number pi is a number h is a constant c is a constant nu 12 is the only thing that you have any control so if nu 12 is large then a by b ratio is large that means spontaneous emission takes preference over stimulated emission so now see you remember when you want to make a laser you want to you want stimulated emission to happen not spontaneous emission so if energy gap is too large then spontaneous emission is favorable so that is not a happy situation for making a laser that is important to understand. So smaller the energy gap better is it is if you are talking about making a laser and that is why the first laser that was ever made was actually a measure not measure ma acr what is measure instead of light it is microwave that is all microwave first laser that was made was in microwave domain and that is where non-linear optics become very important because in most of our applications as we have discussed already you want blue laser you want ultraviolet laser you cannot make them so your only hope is to somehow make a laser in infrared and keep on adding those photons performing second harmonic generation third harmonic generation fourth harmonic generation some frequency generation and produce the light produce light of frequency you require okay this is important to understand smaller the energy gap better chance it has for your spontaneous emission okay and to establish this point I want to do some calculation and as you know in calculations I often go wrong so you better be careful and do the calculation yourself before starting since I already goofed once on the speed of light I do not want to eliminate speed of light and do that what we will use is lambda nu equal to C we know that so nu by C is equal to lambda right so we can substitute and write this in terms of lambda in fact I will go one step forward and write 1 by nu bar for the simple reason that I remember the wave number of microwave I do not remember the wavelength you can do it in whichever way you like but the point is you can use any one of these so you can write a by b in terms of frequency or wavelength or wave number or whatever energy anything see we are trying to make a laser laser that S is for stimulated emission so if you want to make a laser then we have to promote stimulated emission and not spontaneous emission so it is better to work in smaller energy gap if I have said spontaneous I made a mistake are you clear now okay sorry for confusing now let us write this here so what I get is a by b is equal to 8 pi h divided by nu bar cube is that right right so if I write a by b for some nu bar 1 so I will write for nu bar equal to or maybe I will write it in terms of b by a because the thing is you want to promote b we want to promote stimulated emission why keep it in the denominator better keep it in the numerator so we write something like this b by a is equal to nu bar cube divided by 8 pi h and even if I missed out some constant in the process it does not matter so let us see what will be the b by a ratio for nu bar equal to say I am making a mistake here am I not I am surely making a mistake here is it not this cannot be because nu and nu bar would be in the same place so let us see where I have gone wrong lambda nu equal to c so nu by c equal to lambda and this equal to nu bar cube so why have I written it here if this should have come here okay so in fact b by a is equal to 8 pi h divided by nu bar cube are you clear I made a mistake here so I will just go back once a by b equal to 8 pi h multiplied by nu 1 2 by c whole cube so now then I said I do not want lambda I do not want nu I want nu bar so I said lambda nu equal to c therefore nu by c which is a quantity here that is equal to 1 by lambda which is equal to nu bar alright so as I told you already please be careful and correct me wherever I go wrong otherwise we will learn things that are reciprocal of the correct picture okay so b by a is equal to 8 pi h divided by nu bar cube so b by a for nu bar equal to say 200 centimeter inverse divided by b by a for nu bar equal to 20,000 centimeter inverse b by a equal to everything is in the denominator yes man this is a mess 8 pi h so that is right so because here it is 8 pi h nu bar cubed but well it does not matter because I am going to take ratio of ratios okay what is this going to be 200 centimeter inverse what is it is it microwave is it IR is it terahertz yeah what is 200 centimeter inverse what kind of light is it definitely not visible is 200 centimeter inverse 20,000 centimeter inverse what is yeah blue or UV and this is well something that is not exactly microwave it is still IR okay so let us take the now I think the answer is very clear right I will put cube and here I write 200 here I write 20,000 what is the answer 10 to the power 6 right so if you go if you want to make a laser whose wave number is 200 centimeter inverse and if you want to make a laser whose wave number is 20,000 centimeter inverse which one will be easier definitely the 200 centimeter inverse one because the b by a ratio there is 10 to the power 6 times that of the b by a ratio for 20,000 centimeter inverse okay if you just want to work out the value of a value of b it might be a little confusing so b by a ratio is something that allows us to understand what exactly is the situation right are you clear so the take home message here is first of all a is never equal to 0 second one is for laser smaller energy gap is easier so this discussion of two level system has actually given us some very important information what we learnt in the last module is we learnt how to get a relationship between a and b and what we have done so far in this module is that we have seen what is the b by a ratio for different kinds of energy gaps so now let us ask the question what is the finally I want to make a laser right so I want stimulated emission as you said as well as you saw when light comes in and is impinged two level system then you can have either upward transition or downward transition so if you want a laser what you want is more photons should come out then what is going in otherwise how will you have light amplification right so in order to have net stimulated emission the rate of downward transition must be greater than the rate of upward transition I am not writing one two anymore rate of maybe here I should write okay left hand side is rate of downward transition N2 right hand side is rate of upward transition because it has N1 and we have already established that B's are the same energy density is the same anyway so finally what it boils down to is N2 has to be greater than N1 and since we are talking about a two level system we can write N total is equal to N1 plus N2 where N total is the total number of molecules that are there so molecules can either be in state 1 or in state 2 so when you add up the populations of molecules in state 1 and molecules in state 2 you should get N total so another way of writing N2 should be greater than N1 is N2 should be greater than half of N total right and if I put it in words then what we will say is that a population inversion is an essential condition so that we get more light coming out than going in so population inversion is an essential condition without which you cannot get stimulated emission and therefore you cannot get laser so now the question is is it possible to get population inversion in a two level system that is what we will prove I think most of us know the answer that is not possible but we will go ahead and prove it now what I will do is I will write the rate equation what is this term rate of which process B rho12 of nu into N1 it is induced absorption next term I will write is B rho12 of nu N2 what is this stimulated emission okay does it populate or depopulate energy level 2 depopulates so what will be the sign plus or minus the first one populates N2 so sign will be plus second one depopulates N2 so sign will be minus what is the third term spontaneous emission and it will also have minus because it depopulates depletes energy level 2 minus a N2 right now here of course there are well two things to understand is dN2 dt equal to 0 only at steady state but if you start see if you have a situation where initially you have nothing like the light is switched on then until saturation is reached the N1 N2 values will change N1 will keep decreasing N2 will keep increasing okay now it remains to be seen saturation of value of N2 has to be there because you are not producing molecules right so best case scenario is if at all possible all molecules will go to N2 state I am not saying it is possible but it is I am not saying it happens but that would be the best case scenario right initially when you start everything is at N1 no N2 and then after some time all molecules are in the higher energy level that is the best kind of population inversion you could have of course as we will see that is not achievable as you have said already even population inversion is not possible I mean dN2 is never half of N but then the point I am trying to make is that it is dN2 dt dt is not 0 until that steady state is achieved and right now we are talking about the non steady state situation okay now here of course N2 changes with time N1 also changes with time N1 decreases from the initial value and N2 increases from 0 so 2 variables one equation will be difficult but then we know already that I can write like this N1 is equal to N total minus N2 in this equation which are the time dependent quantities is N total time dependent no it is a constant N1 and N2 are time dependent right they are variables but then of course they are correlated to each other so what I can do very conveniently is that I can write I can replace N1 by N total minus N2 that way it becomes an equation in N2 and we are in a comfortable situation we can try to solve it. So let us write quickly rest of it is quite simple algebra and very little calculus dN2 dt equal to b rho 12 of nu N total minus N2. Minus b rho 12 of nu N2 minus A into N2 okay to formulate it makes sense to collect all the terms in N2 in fact out of all the terms that you get only one does not have N2 is not it the first one so that is the only term b rho 12 of nu multiplied by N total is there a constant with respect to time this term that is a constant right and all the other terms are variable not only that the only variable there is N2 and everything has minus in front so I can happily take this minus out and I will take N2 out at the end so what do I get I will write A first and then see this one is b into rho into N2 with the minus sign this one is also b into rho into N2 with the minus sign so it will be A plus 2b rho 12 of nu into N2 right if you are good in calculus you can actually write the solution right away but since I am not good anymore I will do it step by step let me say y equal to right hand side b rho 12 of nu N total minus A plus 2b rho 12 of nu into N2 so what is then dy dt first term is constant N total minus A plus 2b rho 12 of nu into N2 so what is then dy dt first term is constant so that goes second term I can write like this minus A plus 2b rho 12 of nu dN2 dt right so instead of this dN2 dt I can write dy dt divided by minus A plus 2b rho 12 nu and then on the right hand side I can write y is that okay therefore I can write like this dy dt is equal to minus A plus 2b rho 12 nu into y this is dN2 dt equation this one I have said this whole thing to be y is that right now it is not very difficult anymore I hope and I can write something like this dy by y and everybody knows what the integral of dy by y is is equal to minus A plus 2b rho 12 of nu dt what do I do I integrate right only one trick I mean not trick one pitfall is there when I integrate I am going to do a definite integration I am going to integrate from 0 to time t so right hand side is quite simple left hand side is also quite simple it is going to be L and y but we cannot forget what y is upper limit is y I will just write y for the upper limit if you want you can write y dash it does not matter then you have to write t dash here what is the lower limit what is the value of y when t equal to 0 go back to the definition of y at t equal to 0 the second term is 0 because N2 is 0 right there is no molecule in the second level that is what we are working with even though there is a little bit of a problem with that but we can live with it for now we are talking about y so my question is what is the value of y at t equal to 0 second term will be equal to 0 so y will be equal to b into rho 12 of nu multiplied by N total right that is a constant so that is the limit is important b rho 12 of nu N total are you okay this is simple now just write the result left hand side will be ln y I will still keep it at y but then we will change later on divided by b rho 12 of nu N total is equal to minus a plus 2 b rho 12 of nu t see if you are convinced with this next step is very simple raise both sides to find the anti log so right hand side will be e to the power minus a plus 2 b rho 12 into t left hand side nu minus 2 b rho numerator will replace y by b into rho 12 into N total minus a plus 2 b rho 12 into N2 yes but not yet because right now left hand side is in ln right so when I take anti log then left hand side ln will vanish right hand side will be raised to the power of e right so I have not done that yet I am going to do now are you okay with this okay so let me keep it in such a way that you can still see it that is a solution to our problem of course the thing that you cannot see still is what is y need that also so what I can write is now I will try and expand you have understood what I am doing raising both sides right and taking anti log right and let us not forget what is y so what will the left hand side be ln will vanish so instead of y I am writing this expression that is all is a little long not not difficult b into rho 12 of nu N total minus a plus 2 b rho 12 of nu into N2 divided by b rho 12 of nu N total even before I write the right hand side do you see the beauty of this expression if I expand the first term is 1 second term has N2 by N total N2 by N total is what we are looking for right so we are getting there is equal to is equal to N2 by N2 H 2 plus 2 by N total is what we are looking for right so we are getting there this is 1 for e to the power minus a plus 2b rho 12 of nu t. So now simplify a little bit, so left hand side you can see right it is 1 minus this thing divided by this, so can I jump a step, I will just take the second term to right hand side so that minus sign will go, I will take one to the other side, are you okay with that? So then what I have is a plus 2b rho 12 nu N2 divided by b rho 12 nu N total, I have taken this to the right hand side, so on the left hand side I have 1 and I am taking this actually to the left hand side, so minus e to the power minus a plus 2b rho 12 of nu t okay. Now climax is there N2 by N total is equal to b rho 12 nu divided by a plus 2b rho 12 nu multiplied by 1 minus e to the power minus a plus 2b rho 12 of nu t okay. This is the expression for N2 by N total, my job now is to see is it ever possible for this ratio to be more than half okay because if it is, if that happens then population inversion is going to happen, let us see, let me plot, I think we can understand pictures better than numbers, so let me plot N2 versus time, as we have discussed already N2 at time t equal to 0 is 0 okay. What is N2 at time t equal to infinity, this is a monotonic function right, at time t equal to infinity what will happen, this will become 0 the exponential term right because e to the power minus right, so take it in the denominator that will denominator will become a large number, so it is going to be 0 and then you are left with 1, so what we get is that we get a monotonic increase like this to saturation and what is the saturation value I can write N2 at time t tending to infinity, what will that be, well actually N total should be here then, what will that be, what will N2 by N total be, so this thing is gone right, so N2 by N total best possible value, maximum value at time t equal to infinity, turns out to be b into rho divided by a plus 2 into b into rho okay, so this is actually the best possible value, saturation value t tends to infinity of course, when I say infinity I do not mean real infinity, it can be really small time but it is saturation value that is all, now c, now remember we said that a being non-zero is going to be useful, here it is useful, see if a would be equal to 0 then the best then it would have been half right, so population not exactly inversion but at least population equalization would have happened but what you have said is that a is never 0 is b is not equal to 0, if b is not equal to 0, that means that this ratio N2 at time t tends to infinity by N total will always be less than half, have I been able to make the point, so this is why population inversion can never be achieved in a two state system, the best possible value is still a little less than half okay and half is not good enough actually, little more than half is what is required for population inversion that is never achieved, even half is not achieved, so this two level system which has given us all the important quantities that are relevant to lasers unfortunately cannot give us an actual laser, if you want to make a laser the stimulated emission will be there, we have to add at least one or two more levels, so that is what we will study in the next module, we will talk about three level and four level system and then maybe I think we will have time for it, we will do a little bit of calculation on our own laser to get an idea of some of the numbers associated with it.