 So, we are slowly going towards understanding how ultra short pulses are produced and on our way we have discussed what longitudinal modes are whatever discussion we are doing now is completely from this introduction to laser spectroscopy this is the cover of the book it is not very costly 500 rupees now used to be 8000 at one point of time. So, this is what we are going to follow by enlarge for next few modules. So, to remind ourselves what we have learned is that a laser is we always think that a laser is monochromatic, but what we have discussed is that it is strictly speaking most lasers are not because the comprise of longitudinal modes what is the meaning of the longitudinal modes we said that to set up a standing wave the essential condition that has to be fulfilled by a wave is that the cavity length L must be an integral multiple of a half wave length n lambda by 2 equal to L. So, there can be many such modes that are that can be there in the laser as we have discussed earlier we are not really dealing with n equal to 1 2 3 we are dealing with n equal to very large number 10 to the power 6 10 to the power 4. So, 10 to the power 4 plus 1 is almost 10 to the power 4. So, wavelengths of modes that are adjacent to each other do not really vary too much that is the first thing to understand. Secondly, we have worked out the expression for delta nu where delta nu is the reference in frequencies of 2 successive modes mode number n and mode number n plus 1 and that turned out to be C divided by 2L and the C divided by 2L is actually an interesting quantity. What is it? C by 2L or 2L by C what is 2L by C? It is a round trip time. So, it turns out that delta nu the difference between 2 modes 2 successive modes is equal to round trip time inverse of round trip time rather alright. If you do not remember this you better write it down we are going to need this later on in this module or next. And this is another thing that we said that if you think of these spontaneous emission band spontaneous emission spectrum and if you think of stimulated emission spectrum we discussed why stimulated emission spectrum is narrower and we said that let us say that this laser action threshold is somewhere here and we said it is about at half the intensity. We said if the half width at half max is delta lambda then delta lambda is going to be the half width at the base of the stimulated emission band okay. So, now we are really talking about not full width at half maximum but pulse duration which is the time for a pulse to go from 0 through a maximum to 0 okay. Well in this case it is not really time we are talking about lambda but as you know frequency and time are inter convertible by Fourier transformation we are going to invoke it in this module or the next one. This is another thing that we have studied we asked that if this is the spectrum of stimulated emission how many longitudinal modes are there and the answer was capital N equal to approximately 4L delta lambda divided by lambda 0 square. What is lambda 0 square? Lambda 0 is the wavelength where the spectral maximum occurs and what is delta lambda? Half width at half maximum of spontaneous emission. So, it is basically half width at base of stimulated emission band and what was that capital delta nu that we talked about a little while ago delta nu what was the delta nu? Difference between frequencies of two successive nodes. So, please do not confuse that capital delta nu with this small delta lambda. This delta lambda is for the entire spectrum for the entire stimulated emission band that capital delta nu is the difference in frequencies between two successive longitudinal modes. I have this bad habit of saying normal modes all the time. So, if I say normal mode by mistake I really mean longitudinal modes here I do not really mean normal modes of vibration or any such thing. So, this is another important quantity that we are going to use. So, if you do not remember you do not I do not expect you to remember it but better write it down is going to come handy later on. Total number of longitudinal modes is 4L delta lambda divided by lambda 0 square and then one thing that I would like to draw your attention to which we did not say in the last couple of modules is do not think that the longitudinal modes have spectra that look like delta functions. So far what are we saying? We are saying that this nth longitudinal mode has some frequency nu n. The next one n plus 1th longitudinal mode has frequency nu n plus 1. So, that might give you the idea that these are delta functions but that is not true. Each longitudinal mode has a spectrum which has finite width do you agree or do you not agree with this? Let me rephrase the question is it ever possible to have a spectrum which is a delta function in the truest sense? No why not? Why can we never get for any system a spectrum which is a delta function? And here I am talking about energy domain spectrum yes. So, this takes us back to the semi classical treatment once again we keep referring to it again and again in this discussion time dependent perturbation theoretical treatment of interaction of radiation with matter. So, there if you work it out you will see that well of course you have a spectrum there are many line broadening mechanism that can take place some of them are homogeneous some of them are inhomogeneous. We are going to talk about homogeneous and inhomogeneous line broadening mechanisms later in this course. So, you cannot have a course talking about lasers and not about line widths and in the next couple of modules we get sort of get an idea why right. So, can we name some line broadening mechanisms yeah I will tell you one collisional broadening right molecules that collide with each other they transfer energy and so energy is changed and that is why spectral line gets broadened Doppler broadening somebody said Doppler broadening is also correct. So, what is Doppler broadening? It is best understood in terms of emission your system emits light and it goes towards the detector now if the molecule is also moving towards the detector or away from the detector then the effective length changes and effective frequency changes very much like what we have studied in class 11 and 12 physics where the train comes towards you the whistle sounds more shill more shill means frequency is higher and when the engine goes away from you the it sounds less shill because there is more space in that time and the wave is getting important that is called Doppler broadening. So, Doppler broadening is an important issue in spectrum in spectroscopy but suppose I say that I have a situation where I completely stop collisions you freeze the sample or something nothing is moving and since nothing is moving Doppler broadening also does not arise still there is some mechanism no matter what you do you cannot get rid of one thing that will invariably add to a line width let us say let us have quenched vibration of course I can never quench vibration because it will be in that v equal to 0 state. So, the mechanism that you cannot do anything about is natural broadening or lifetime broadening right it is something like you can if I put it very qualitatively what it means is ground state has a lifetime of infinity let us say and excited state has us has some lifetime. So, of course there will be some uncertainty in time and we know uncertainty principle uncertainty in time multiplied by uncertainty in energy is a constant. So, what is the maximum uncertainty in time allowed associated with the excited state cannot be more than its lifetime see why I say my height is 5 feet plus minus 30 feet does it make any sense it does not if I say my height is 5 feet plus minus 5 feet then also it does not make sense to me for me, but it does make sense for quantum mechanical objects as you can only do so much. So, that is the maximum limit on uncertainty. So, since that uncertainty is limited you will have some uncertainty in energy also you cannot have delta e almost equal to 0. So, speaking very qualitatively that is what gives rise that is why it is called lifetime broadening ok. So, some broadening will be there ok. So, it is important to recognize that each mode each longitudinal mode also has a spectrum that is not a delta function at least natural line broadening will be there let us say that is delta less my problem is I have used such a color that I myself cannot treat anyway. So, delta less spectral width of each mode it is important to recognize that it is there ok. Now, next what I will do is I will take a little bit of a detour we will come back to this mode locking business mode locking is a technique you want to discuss today by which you can produce ultrafast pulses, but there is another method by which you cannot perhaps produce an ultrafast pulse, but you can produce nanosecond pulses also it is called Q switching you are right. So, what is this Q where did it come from is it James Bond Q or what is it. So, most likely we will have to come back to this issue of Q, but let me for the next 3, 4 minutes at least provide a glimpse of what it is Q is quality factor Q for quality and it is defined like this 2 pi multiplied by energy gained in the system divided by energy lost during one cycle ok. Why will energy gain be there in a laser I am talking about laser of course. What is the mechanism by which energy gain can take place and when I talk about Q I am talking about the cavity ok. So, what happens is you go back to the very basics of lasers you have this active medium that is giving out light right and it is doing round trips in every round trip the number of photons is getting increased. So, in other words energy is it increasing right that is called gain and how do I get lost I mean I do not get lost how do I lose energy how does the cavity lose energy. One thing you can think is suppose now light goes out of the cavity that is actually a loss ok that is what we want, but as far as the system is concerned it is a loss, but for now let us not even talk about light going out from there. If you even put 200 percent reflecting mirrors will it just keep on gaining energy or will there be some there will invariably be some loss why because system will get heated up heating is a very big problem in laser. So, you see you have to use things like chillers and all many times when you use powerful lasers right there can be diffraction there can be scattering. So, there can be many mechanisms. So, this Q is given by 2 pi into energy gain and system by energy lost and when I say Q switching what do I mean then I mean that for some time I let the Q go up right that means energy is getting built in the system then by some mechanism I switch the energy out. So, what will happen at that instant Q will fall right. So, Q will switch from a very high value to very very low value ok then again Q will start building up. So, if you can somehow achieve this that is one way of producing pulse laser Q switching. How we achieve this we are going to discuss when we talk about the actual instrumentation part alright, but the point is since there is some loss it is no longer fair to think of this wave that I have as just a plane wave right is going to be sort of a damped oscillation if there is a loss right. So, it will be something like this do not worry too much the purpose is not to do every bit of math that is associated but rather to get working idea ok. So, we will take things axiomatically, but these are not all that difficult also. So, something like this is an oscillation and there is a decay and this figure here is not really to scale it is not to scale because this T0 that you see T0 is basically the time difference between two maxima time interval between two successive maxima and tau is ok time for decay of amplitude from A0 to A0 by E does that ring a bell sort of something like lifetime constant right. Now, generally T0 is actually much much smaller than tau otherwise you are very bad laser alright. If it decays completely before even before doing a cycle then how will you work. So, this figure is just is not to scale is drawn in such a way so that you can see all the quantities involved that is all. But T0 is actually much much smaller than tau and we are going to use it shortly. But let me write something let me write this 2 pi then what is energy gain in the system A0 square what is energy lost in one cycle it will be A0 square minus A0 square into e to the power minus 2 beta by T0 for now do not worry about what beta is we will have to come back to that we can go to the final expression A0 square will cancel between numerator denominator you are left with 2 pi divided by 1 minus e to the power minus 2 T0 by tau which is a very familiar form of equation for us right ok. Now we can simplify this expression a little bit precisely because T0 is so much smaller than tau. So, what will happen if T0 is much smaller than tau how do I expand e to the power minus 2 T0 by tau this is a technique that we have used in almost all physical chemistry courses right if you do quantum mechanics statistical mechanics whatever you always use this kind of an approximation T0 is much much smaller than tau. So, how do I expand e to the power minus 2 T0 by tau e to the power minus x when x is very small 1 1 is correct yes. So, I can write something like this of course there are higher terms but as it always happens higher terms are we are saying that this itself is very small. So, higher terms will be even smaller. So, we neglect them. So, we can just put e to the power minus 2 T0 by tau to be equal to 1 minus 2 T0 by tau and if I take this expression and plug it in into the expression of Q what will I get very convenient one and one will cancel each other what am I left with yeah something like this ok. Now, so 2 and 2 will cancel of course 1 by T0 what is T0 is time period right what is 1 by T0 1 by time period frequency naturally and of course the answer is also in front of you pi nu 0 tau ok and then pi nu 0 is the angular frequency we are going to use angular frequency time and again here because we are dealing with periodic functions they are going to repeat after regular intervals. So, angular frequency very often turns out to be an easier parameter to use. So, you get omega 0 tau by 2 ok. Now, next part I am not going to derive I will just tell you that delta less what is delta less spectral width of each mode is given by 2 pi into omega 0 by Q ok 2 pi into omega 0 by Q. So, this is the expression for spectral width of each mode the purpose of this discussion is 2 fold first to emphasize the fact that longitudinal modes do not have delta function spectra second to introduce this important parameter quality factor Q which plays a very important role in producing larger pulses and also as we will see later on the same kind of device is used when we want to amplify a laser. So, Q factor is something that will come back again and again. Now, let us come back to our discussion where we had stopped in the last module we said when we take longitudinal modes in a bunch what do you expect to get and we have talked about 2 situations one in which delta phi is constant what is delta phi phase difference between 2 successive longitudinal modes constant means it is independent of time and second one where delta phi is a function of time. So, let us take the second one first if phase difference is a function of time then you have a very chaotic situation right there is no correlation now it is there now it is not there. So, you are going to get something like this ok you are going to get a free running laser with inherent fluctuations that might be there. However, if delta phi is constant then what happens we did not we discussed it in the last module but we did not really spend too much of time on it. So, for the benefit of those who might be new to something like this it is not very difficult to understand first of all it is understood quite is simply by using simple day to day analogs ok think of a team of runners 5 runners and they are running around a field running around gymkhana they start together when they start together they are in phase ok when they start running somebody runs faster somebody runs slower what will happen there will be a spread right and then it is possible that they are going to come back all together if they keep on running at the same speed for a long long time there will be times when they come together there will be times when they move away from each other it might be difficult to believe if I am talking about 10 runners but it is absolutely easy to understand if I talk about 2 right suppose Shorodip and I runner race around gymkhana ground what will happen of course he will win as he can run faster he will not win ok. So, let us say Tanuja and I runner this Tanuja will definitely win right so what will happen we will start off together then we are in phase right and then what happens is Tanuja starts taking a lead and I start lagging and then if our speeds are sufficiently different what will happen eventually she will catch up I will have done 5 laps she would have done 6 and she would have caught up with me ok. So, that time we are again in phase whenever we come together we are in phase when we start moving away from each other defacing starts and when we are diametrically opposite in the field that is absolute destructive interference right. So, now just extend this to many runners that is exactly what happens when delta phi is constant well what I discussed is when delta phi becomes 0 delta phi equal to 0 is easier to understand perhaps what will happen is at this time let us say all the waves are in phase all the maximum match then what will happen x axis is time they are different frequencies. So, when you go away from here some of these amplitudes will decay slower some of these will decay faster ok. So, phase difference is there but then what will happen is you do not have constructive interference anymore ok. So, if you look at the resultant amplitude that is going to fall it might do some beats up and down and then it will become 0 after some time they will come in come together once again ok. This is something that you encounter very frequently when you discuss things like NMR spectroscopy in time domain then you talk about interferograms. So, the important thing here that we are going to use is y axis is amplitude what is the relationship between amplitude and intensity what is the relationship between amplitude and intensity actually mod square field does have an imaginary component ok. So, mod square if I take mod square of something like this what will I get everything will be positive right will become sharper. So, we get something like this ok and as we are going to see in the next module we do not exactly get something like this we get a little more structure in each of the pulses. So, but the crux of the matter is you get pulsed operation and we are going to show in the next module that separation between two pulses is going to be 2L by C. So, what I have written this may be a little confusing did I see the opposite ok. A star means complex conjugate. So, separation is going to be 2L by C round trip time which is great this is why we can do mode locking actually you know separation between pulses is actually round trip time. And then pulse duration as we are going to show in the next module is 2L divided by Cn what is the meaning of N capital N number of longitudinal modes that are locked. This is what brings us to mode locking if you can lock a large number of modes then we get short pulses because see N comes in the denominator if N is large then TP the pulse duration is going to be small if N is small then pulse duration is going to be large. So, what do you need do you need a single mode laser or do you need a multi-mode laser you need a multi-mode laser. What is the meaning of a multi-mode laser if I put it in simpler terms the bandwidth should be large right when I say bandwidth here I basically mean the spectrum should be as broad as possible if the spectrum is too narrow then that will mean you will have lesser number of longitudinal modes there. So, that is one essential condition that is why it is not so easy to make a an ultrafast laser using a gas as an active medium because it has very sharp lines you need something that has a broad spectral band otherwise it will just not work and so we see that that related to each other an ultra short laser is always going to be a broader band laser. So, if you we are going to go back to the lab and we are going to demonstrate the way we know that our laser has become pulsed is that we are going to discuss how you can measure the pulse width also using something called autocorrelator but we do not even do that all we do is we look at the spectrum if the spectrum is very narrow then we know that it is not pulsed if the spectrum is broad then we know it is pulsed right. So, that so in the next module which is going to be perhaps a little smaller than what it you what our modules usually are we are going to sort of derive these expressions even though t equal to 2L by C might sound an obvious conclusion but we will still derive it but while deriving it we are going to jump steps we are going to take things axiomatically the idea is we want an overall picture of why things are how they are and another thing that we learn is that this the way we have drawn it here is actually a simplification you get a picture like this when you lock a small number of modes when you lock a large number of modes you do not get something you get something that looks a little different that is what we are going to discuss.