 We have learnt in the previous modules that this two level system that is so useful in developing the concept of stimulated emission and spontaneous emission is actually useless as far as making lasers in real life are concerned. So what we will see in this module is what happens when we have more levels than 2. But before going there let me ask a question about two level system. See the way we introduced it was that when you perform the semi-classical treatment using perturbation theory time dependent perturbation theory then you get an idea about absorption and stimulated emission and then we said that it was a genius of Einstein that brought in the continuous emission term in the kinetic equations he wrote because it is there it is real. But I hope you have not forgotten that we also made a passing comment that even then the picture is not complete. Can you tell me why we made that statement? Why is the picture not complete for even a two state system? Even if you consider absorption, stimulated emission and spontaneous emission. What have we missed out on? Yes, so we have missed out on something that is actually the focus of our research entirely. We have missed out on the non-radiative relaxation. Even in Einstein's treatment non-radiative relaxation was not there. So that comes in as another yet another term in the kinetic equations and when you bring it in then you get all those relations we have written earlier the rate constant of non-radiative processes quantum yield and lifetime they are related right 1-5f divided by tau f. So that is where that comes from but in the discussion that we are doing so far we are pretending as if there is no non-radiative relaxation which of course is not correct and it does not give you a complete picture. But in the discussion that we are going to do now we are going to invoke rather non-radiative processes to understand the situation a little bit qualitatively. I mean one can go ahead and do a lot more kinetic equations and convince ourselves what the situation really is for a 3 level and 4 level system. But once you have done 2 level system rest of it is just an extension you can do it yourself. So we will not try to do it in the course I will draw the picture perform a qualitative discussion and then move further ahead okay. So 2 level system as we said is not good enough to give you net stimulated emission okay. What about a 3 level system suppose I have another energy level 3 and as said we are going to perform a qualitative description here and the situation is this that you promote the molecule from energy level 1 to energy level 2 and let us say that there is some efficient non-radiative relaxation pathway that quickly brings a system down to level number 3. If you read textbooks on laser in many cases they have not used the discussion of non-radiative pathway they have stuck to discussion of radiative pathway and it might as well be there may be a radiative transition from 2 to 3 the only condition is that it has to be fast then what happens. And let us say so an example that we will all understand is let us say that state 1 is S0 ground singlet state, state 2 is S1 excited singlet state, state 3 is a triplet state. So in that case this non-radiative process would be ISC and as you know ISC can actually be promoted by using heavy atoms and all okay. There are mechanisms by which you can actually make ISC very efficient of course purists would say that in that case the state number 3 is no longer a triplet because when you invoke a heavy atom effect then what you have is you have spin orbit coupling and it is no longer a pure triplet state. But that is fine I mean we can still go ahead with the fact remains that it is not so easy for the system to come back from 3 to 1 anyway. So as we know triplets have a long lifetime which is sort of a quasi-permanent state right then what will happen? Let us say that this process takes place in a few femtoseconds or few picosecond let us say and let us say that this state number 3 has a lifetime of let us say microsecond microsecond is enough how many picoseconds are there in a microsecond 2 picoseconds are there in one microsecond please come back to the linear world yeah so can we try again how many picoseconds are there in a microsecond 10 to the power 6 right 10 to the power 6. So what I am trying to say is that before a molecule can come down from 3 to 1 this is the lifetimes a large population of 3 can actually be built up if this process 2 to 3 radiative or non-radiative whatever takes place in picosecond time scale and if lifetime of this is microsecond also then before a molecule that goes here can come down to the ground state this 2 to 3 conversion will take place may be 10 to the power 6 times and 10 to the power 6 is a large number. So what is happening for a sufficiently long lifetime of this state number 3 is population will build up N3 and N1 will keep decreasing okay is that right? So what you are doing essentially is that you are pumping population from state 1 to state 3 through state 2 and if this 2 to 3 relaxation is fast enough and if state 3 is sufficiently long lived then you can achieve a situation can reach a situation where N3 can become greater than N1 and then you can get spontaneous emission between 3 and 1 is that right? Have you understood what is going on? So for what we have discussed now N2 is populated from N1 we can say instantaneously time for transition is at a second from 2 to 3 that population takes place very efficiently okay from 2 second process let us say what we are saying is 3 is very very long lived if all the numbers work out nicely and that is you can get an actual feel of the numbers if you work out the differential equations then you can hope to achieve population inversion between 3 and 1 and once that is achieved you can have lazy as everybody understood and that brings us to something that is very very relevant to our core discussion there are a pulse laser because you see as long as so initially think of time 0 population of state 3 level 3 is 0 population of level 1 is almost in total. Now think of say 1 nanosecond after you have started light is shining right the pumping light 1 to 2 that is shining what N3 is building up but still perhaps it has not reached population inversion so for the first 1 nanosecond there is no lazy for 2 nanosecond also maybe there is no lazy maybe after 500 nanosecond or 1 microsecond or 5 microsecond provided lifetime of 3 is long enough population inversion will be achieved. So if you think of the output of the laser there is no lasing for a long time right but when population inversion is achieved this is the important part then all the population will want to get depleted by stimulated emission at 1 go right so the output will get will be something like this and then again if this 1 to 2 radiation is on population of N3 will keep building up and the laser will be silent x axis is time y axis we can say is intensity of keep going then after the required amount of time once again there will be a burst of light and this will go on provided you keep the pumping on. So for a 3 level system like this where 2 to 3 transition is ultrafast and 3 is ultra long lived you are going to get a pulsed output case in point ruby laser so this is one situation now we will come back to level 3 also level 3 once again but let us talk about 4 level system first. Let us say I have a situation like this 1 2 then let us say there is 3 and 4 and situation is like this direct transition is possible between 1 and 2 from 2 to 3 you have an efficient non-radiative pathway from 4 to 1 also you have an efficient non-radiative pathway can you actually have a system like this well suppose you have intermolecular proton transfer excited state intermolecular proton transfer right then it is possible then if you look at the energy profile then most likely it is going to be like this okay so I will draw a schematic molecule from OH is there and nitrogen atom is there not even drawing the rest of the molecule here so what I am saying is this is the ground state corresponding to it then you excite and excited state proton gets transferred to nitrogen excited state intermolecular proton transfer. So even before going there if ground state this is the situation what I am saying is that nitrogen being protonated that is not an energetically favorable situation so that is when you have this asymmetric double well potential y axis is potential energy right what is x axis you can say bond length for now or reaction coordinate so you have asymmetric double well potential what happens when you excite as we might have discussed in this course acids are more acidic bases are more basic in the excited state OH is an acidic group n is a basic group the excited state due to change in acidity and basicity proton gets transferred so now this locally excited state for this one is no longer energetically stable so when you excite then you are going to reach a state that is actually energetically less stable so excited state potential energy surface you can expect is something like this this state is OH double bonded to n this excited state is also OH not double bonded hydrogen bonded to n then proton transfer has taken place this is hydrogen is covalently bonded to n hydrogen bonded to oxygen this is the corresponding ground state okay so if proton transfer is sufficiently facile what will happen you excite immediately proton transfer will take place and then whatever emission takes place will be from here to here but then since this is energetically not favorable the moment it has any population it will come back to this 4 level system so that is why ESIPT molecules have been touted for a long time as good candidates for making lasers because they are actually 4 level system but now see what kind of output we will get let us write when you start population of n1 of 1 is n1 what is population of 2 0 this is 0 this is 0 a little while later so n1 equal to n total you start from that a little while later what happens this decreases a little bit population of n2 is still 0 because there is an efficient non radiative pathway to level number 3 all right level number 3 let us say has some life time whatever so n3 population is there what about n4 what we are saying is if you look at this diagram the moment level 4 is populated it goes back to 1 so n4 population is maintained to close to 0 at all times is important to understand here if the rate constants are such that n4 concentration is held to be near 0 at all times then what will happen population inversion is already achieved between third level and fourth level because even if there is like 2 molecules in n3 population of fourth level is 0 right so in this case what kind of an output you will get there might be an initial induction time but if you neglect that you get a continuous output so this is an example of an the first one third level 3 level system is an example of an intrinsically pulsed laser this is an example of an intrinsically continuous wave laser so I told you an example of a 3 level system that is actually something like this does anybody know an example of a 4 level system something is very very common and not necessarily in ultrafast but I do not exactly know it must be a diode laser so yes most likely yes ndi laser is what I had in my mind ndi laser is perhaps the most celebrated example of a 4 level laser so it is supposed to be continuous right so you see that ndi lasers are there that arc that have continuous wave output and then you see ndi lasers that are pulsed so trying to say is ndi lasers are actually intrinsically not pulsed intrinsically they are cw lasers if you want to make them pulsed if you want to get pulses out of them then you have to do something yourself and the doing something yourself is either q switching or mode locking so that is what we are going to learn in the subsequent modules may be not very sure whether it will be exactly in the next module you have not made up my mind yet how much I want to talk about gain and loss and q factor perhaps you should do a little bit but eventually we are going to talk about mode locking okay so remember something that is intrinsically cw can per force we made to give you a pulsed output okay and it is not very difficult to understand as well the light coming from that lamp is cw right and the detector I have is my eye so I keep it shut I do not see it then I open it for a small time and close it again so that way the detector that I have actually gets pulsed light out of a cw source right or I could use a shutter well I lead is a shutter okay but here I cheated a little bit because I talked about shuttering the detector rather than the laser itself what I can do is maybe I can keep it off for some time turn on the switch only for a short time periodically that may not be a very good thing to do because it might spoil it but there are other ways we can use different kinds of shutters by which a cw light can be made pulsed and that is what we will learn in detail because that is that is really something that is very important to our course of study but before we leave this discussion let me talk about another kind of three level system I say another kind three level system has to have three levels cannot be there what I could cannot really do much about 2 to 3 also but suppose now I have something in which this 2 to 3 conversion is radiative and 2 has a long lifetime 3 to 1 is non-radiative and fast of course there is a catch here the catch is what about 2 to 1 emission will come to that but for the moment you tell me if this is the situation then what will happen what kind of laser do I get first of all this is where the leasing will take place right between 2 and 3 will it be cw will it be pulsed it will be cw so you can have three level systems in which the output is continuous wave it is not necessary that whenever you have three level system the output is going to be pulsed it all depends on the relative rate constants okay but now who will take care of the 2 to 1 1 to 2 absorption is taking place so 2 to 1 can also take place and I am trying to have induced emission. So do I have some way in which I can force the system to give me leasing in 2 to 3 and not 2 to 1 yes so energy density matters right do not forget energy density is another factor here so if energy density of nu 2 1 is very low and energy density of 2 to 3 is high you introduce some photons from somewhere of that energy then you can make a laser emitting in that particular frequency and this is actually discussed in Macquarie and Simons book in my edition the page number is 603 here they have talked about a three level system and you see they have considered all kinds of if you look carefully there is no non-radiative transition here it is radiative all the way but they have said the conditions in such a way that you have a three level system where the leasing is between 3 and 2 and the output is continuous okay so I would advise you to try and work out this problems you see finally you get an expression of N3 by N2 and so on and so forth so do it yourself and satisfy yourselves that this is actually the case okay we will move on a little bit and give you a quick glimpse of what is to come okay and let me show you since I have the book open instead of drawing let me just show you the diagram this is what you have inside a laser I think this is something that all of us know how do you make a laser how do you get amplification what to do is first of all you need some pump source you have to create that appropriate excited state pumping can be done either optically or electrically you can have some light source so if you see older NDAG lasers they use flash lamp to excite so in the laser that I used as a PhD student had this NDAG rod about this big and on two sides there are two flash lamps the arrangement was like this parallel you can think that the blue pen you better look at the projection then you will see blue pen is the agrod and let us say the black pen is a flash lamp and you have another flash lamp on the other side and we had this parabolic mirror so light from here would be focused on to the agrod of course 25 30 years ago things were much more primitive than what they are today and there is always this problem of heating you have this flash lamp on all the time and it is pumping so things would get heated how do you dissipate the heat there was this actually very nice if a little cumbersome design where this whole thing was immersed in a tub of water a tub is about this big the whole thing was in water so of course that water had to be distilled water and then constant temperature had to be maintained so there was secondary cooling and all but you know if you use a bulb sometime or the other it goes bad and if you are unlucky this flash lamps instead of just dying on you would burst and then whether the burst or not to change the flash lamp we had to take out 12 13 screws drain the water completely then salvage broken pieces or the unbroken flash lamp from there and change it so those were more interesting days so that is one way of pumping what we do now in a diode pump solid state lasers is that you have a diode bank which gives you light which is directed by photo diode on to the ag laser and whoever has used that laser would know that there is a chiller that provides cooling but things are not as messy as they used to be earlier now it is not necessary that you always do optical pump you can do electrical pumping as well but let me decide whether we want to talk more about that later and then you have this I do not know if you can read the projection here then you have this gain medium gain medium or active medium that is the molecule or material or iron which actually gives the emission so if you are talking about ruby laser your gain medium contains ruby what is ruby we will come to that a little later I will show you if you can read and then what you do is alright em emission takes place and that emission photon that are there you have two mirrors on two sides so for photons that go to this mirror and typically you have curved mirror because you want to focus here so that you had density of photons so let us think of one photon which has gone here hit this mirror come back and when this photon passes through the gain medium is going to experience other ions or other molecules or other material that is excited already by the pump source you get my point first think of it like this pump source let us think of it as excitation light 1 to 2 excitation okay I have done that and some emission has taken place let us say the stimulator is spontaneous emission does not matter it can take place in all directions but we are talking about this mirror so a photon that goes and hits this mirror comes back it traces its path now when the photon comes back to the gain medium it is going to find other molecules or other ions whatever the gain medium is in excited state who has taken them to excited state the pump source is it right it is a pump source that is exciting the components of the gain medium the job of the photon that is initially emitted and has come back is to cause either promotion of unexcited molecules or deexcitation of already excited molecules this finds both okay and that is where this discussion of gain and loss and all that comes in maybe we will do it later so think of this photon comes depopulates one ion or one atom or one molecule in the excited state through stimulated emission the important thing to understand is what we had drawn at the beginning of the 1817 16th whatever it is called 16th module 16th lecture remember the photon that comes in is not absorbed it goes out itself but it deexcites a molecule and causes the emission of a second photon so now you can think like this photon goes out by whatever mechanism of emission comes back causes stimulated emission now 2 photons come here hit this mirror come back and they cause 2 more photons to come out so in every round trip the number sort of doubles in the best case scenario it does not always double it will always be less than 2 actually the factor but it increases okay so in a very short time okay how much time does a round trip take let us say let us do a simple calculation let us say this length is 1 meter so round trip would take how much start from here okay go here go here and come back let us say complete round trip it is travelling 2 meter in that case okay let us say 3 meter let us say 1.5 is the cavity length then it is easier then round trip distance is 3 meter what is the speed of light 3 into 10 to the power 8 meters per second this time we have not made a mistake so you can calculate how much time it takes very small right so if you are talking about some seconds in a few seconds there is a huge amount huge number of photons that is produced okay and that is called gain increase in number of photons but now what is the point I mean see if you are a miser you work very hard earn a lot of money keep everything in an iron box and then when your time in the earth is over of what use is that money to you right you have to spend it also so if you are going to make a laser it is not enough we just have a lot of gain you have also need to have a little bit of loss I mean light should come out and the easiest way in which it is done is that on one side you have a high reflective mirror reflectance 100% on the other side you have a partially reflecting mirror so say 95 99% reflecting mirror 1% will come out okay this is called the output coupler we will discuss these terms again in more detail okay then what will happen so initially even number of photons is small you hardly see any output because say 1% of light is coming out let us say I am being a little strange and tear but when there is a lot of gain there is a build up then 1% of that amount is also good enough then you get laser light coming out from this side okay this is the simplest way in which you can depict a laser now we will get into more complication what happens what should a cavity length be I arbitrarily said one and then I said 1.5 is it arbitrary can I have any laser light even for CW and if I want pulsed light what is the most stringent condition that comes these are things that will slowly get into but let me complete this discussion by showing you another page of Macquarie if you cannot read here please go back and read in your home this table 15.2 in Macquarie and Simon provides you a compendium of some solid state lasers and what you see is chromium so it is important to understand what this is you always say ND egg laser and forget about it it is important to remember what is the role of what the role of ND is and what is what the role of Yag is what is Yag it is aluminum garnet and what is that it is actually AL 5 O 15 okay Y 3 AL 5 O 5 have you heard the word garnet in any other context is a precious stone jewelry yes have you heard of ruby in some other context of course you have okay so all these precious stones are precious to us not because you want to wear earrings I mean not all of us but because they actually make good gain media but it is important to understand that it is the dopant the iron that is actually the emissive species the role of Yag or Yale for glass all that is to provide a support in fact when you talk about ruby laser natural ruby does not give you good laser ruby laser invariably has synthetic ruby which is grown so that the crystal structure is good and natural ruby is not very easy to make use of as a laser material so here the last example that discussed is our good old titanium sapphire laser here this Ti 3 plus iron that is what gives you the emission okay and sapphire again another precious stone is Al 2 O 3 that is the matrix both are important and titanium in some other matrix may not be as stable as titanium sapphire so good thing about why are precious stones precious and let us say I make stone of sodium chloride crystal of sodium chloride and to call it a stone right it is not going to be very precious or calcium carbonate that is actually valid stone why is it that calcium carbonate is not so precious as ruby because the answer is same as if I ask you why is gold precious because gold does not react you make very nice jewelry out of iron then eventually they rust and it is not there anymore in a silver gets black but not as bad as iron gold does not react that is why gold is so precious platinum is precious so here also the good this precious stones are precious because they are so stable they do not react to the slightest moisture or some other gas that might be present that is why you want to make these as support material this is what your titanium sapphire laser actually is and you can also have gases as gain media you can Hini laser is something that you might have heard before the advent of diode lasers Hini lasers were ubiquitous especially where alignment was involved now in many cases they have been replaced by diode lasers because now you have good quality diode lasers which give you nice transverse mode and all okay but that will be discussed sometime later the point I am trying to make here is that your gain medium can be many different things can be solid can be liquid can be gas but it has to satisfy certain properties 3 level or 4 level at least and stability is an issue do you know of any laser where the gain medium is liquid well when I say liquid I mean a solution go to the next lab you will see die laser so what they do is they dissolve something like Rodham in 6g in methanol and keep it in qubits excite using a green laser and make a laser out of it again as PhD student I use what is called a synchronously pumped die laser we will discuss what synchronous pumping is there we used to flow the die we wanted picosecond resolution we cannot use a cuvette so you actually flow the die and you have a jet of die coming out and it is caught by a what is called a catcher tube and that takes it back to the reservoir and the green laser was focused on to that jet of the die like no medium so it does not become broad that was also Rodham in 6g but not dissolved in methanol it dissolved in some viscous solvent so that the jet is good okay so we will come to those one by one I wanted to do a calculation today but maybe we will leave it for a little later when we actually discussed isophire laser right so next day onwards we talk a little more about lasers and finally we are going to arrive at pulse lasers.