 this is the figure I was missing in the morning that. So, what I had tried to point out is this that supposing a beam up. So, this is your picture of an optical fiber you have a core with a higher refractive index than the cladding and of course, you have air outside which has refractive index of 1. And so, supposing there is an angle I at which it is incident and then it gets refracted and then comes here this angle I have not denoted in the picture, but I had written it as theta and then it comes and hits the other surface interface at an angle phi. So, obviously this phi is nothing, but 90 minus theta here. So, the point was that there is a value maximum value of the incident angle for which this process of internal reflection will take place and internal reflection is the primary thing which we worked out in the morning, but I will go through this again. So, this is essentially what we say sin i by sin r equal to n 1 by n 0 instead of r I have used a theta there. And for total reflection the angle of incidence at the interface between the core and the cladding must be greater than the relative refractive index of the second medium with respect to the first, but so sin phi should be greater than n 2 by n 1 which means cos theta should be greater than n 2 by n 1 or sin phi should be greater than n 2 by n 1 that gives us a relationship of this type that this leads to if you use along with this relation sin i should be less than this and I normally take n 0 to be equal to 1. So, therefore, the maximum angle of incidence at which total reflection will occur at the core cladding interface is given by sin i m equal to root of n 1 square minus n 2 square. So, this is this quantity root of n 1 square minus n 2 square is what we call as a numerical aperture. So, numerical aperture just is a name given to it which stands for the light gathering power of the fiber and it sort of tells you that what is the beam angle through which at which the source beam must strike the interface. So, that the guiding action takes place and I had sort of worked out in the morning that this is the i m is about 14 degrees or so. I will also because of continuity problem I will also go through slightly on repeat a part of the linear superposition that I did. So, what we said is that this is the basic principle of interference and diffraction. We talk about two plane polarized wave the first one has the polarization vector E 1 and this is of course, the space time part this is E 2 and this is space time part. Now, you have to add these two to find out what is the effect at a particular point. So, if you add these two. So, you remember how you do it you have to add this plus this plus the complex conjugate of the same thing. So, multiply the two and remember that this E 1 and E 2 are vectors because these have the directions of the polarization of the electric field. In general when we do our problems in Young's double slit we assume that E 1 is parallel to E 2. So, therefore, we ignore its vector character the moment we write it down. We just write without that vector sign there add it up and then get some result. But if you write it fully then actually the interference term is 2 E 1 dot E 2 times cos theta and this is the trivial algebra to work out that this theta that I have written down is given by this. And this is the interference term 2 E 1 dot E 2 cos theta. Normally the way you look at it in your usual thing is it is 2 E 1 E 2 cos theta which is your Young's double slit pattern. Now, the thing that we are trying to say is this that this of course, since that term is not equal to 0 the net intensity is not just the sum of the intensity between the two of the two sources separately taken. And that is the result why we get fringes or interference patterns. And first I made a few comments, but that was not particularly important. The new thing that I wanted to point out is that if E 1 dot E 2 is equal to 0 irrespective of what is the phase difference between them. Then also the interference term is 0 and E 1 dot E 2 term will be 0 provided their polarizations are perpendicular to each other. And this has been experimentally checked and it even follows that you cannot have an interference between a left circularly polarized light and a right circularly polarized light. So this is what where we stopped last time. Now, since we are talking about interference diffraction then we are interested also in learning a little bit about laser. We need to understand the word coherence. What is this coherence? See first let me sort of explain why the typical light that you get from what we call as the thermal source they are in coherence. See if you realize that a typical thermal source the excitation takes place over a period which is let us say the lifetime of an electron in that state which is about 10 to the power minus 8 seconds. What it means is that an electron gets excited stays in the excited level for some time and it takes only about 10 to the power minus 8 seconds to for it to drop back and once it drops back it of course emits a photon. Now which means a typical thermal source is emitting different photons in intervals of 10 to the power minus 8 seconds. Am I making sense? It is not the first time there is one photon after 10 to the power minus 8 seconds there is one photon another photon. In the expression that second line when you write 2 e 1 dot e 2 cos theta. So since cos theta term has already come so we have I think e 1 e 2 are the magnitude of e 1 you realize your problem but that is not what it is. See this is the problem of a memorizing formula a dot b is a b cos theta. But what is cos theta there? There the cos theta is the angle between a a and b but that is not what is cos theta here. Here I have defined what is theta. See do not just go by a formula which you have heard a dot b is a b cos theta, a dot b is a b cos theta there theta is the angle between a and b ok. I could have called it some other psi delta. What I am trying to tell you is just look at the multiplication here right. I have given the expression for e 1 and e 2 there. Say this is e 1 and mind you what is the difference? This is a vector I have not given you e 1 times something plus e 2 times something. Because this is a vector so it is actually multiplying this vector is a space time factor am I clear? And that is because the electric field expression because I am taking a linearly polarized wave there is not only a space time variation but there is a direction associated. So, to e cos omega t minus phi or plus phi you provide for example a direction i. Now you are taking a dot product of that with the e 2 which may have a different direction. It is the angle between e 1 and e 2 which will appear in that e 1 dot e 2 dot. But there is something else that is happening here when you take i equal to this multiply this you get e 1 dot e 2 star e 1 star which is e 1 square e 2 dot e 2 star which is e 2 square. But there is this thing you see e 1 dot e 2 star plus e 2 dot e 1 star. Now look at what are these? So, e 1 dot e 2 star e 2 dot e 1 star you have these expressions here. So, you will have to write this as this one times exponential of this, this times exponential of minus that. So, it is those trigonometric expression that I have added up and I have defined it to be theta. It is not the angle between e 1 and e 2 am I clear? So, I have a doubt in constructive interference in a wave front many waves present all needs to be in phase. But if particular wave is some phase shift is there can it be allowed to pass through the core the resultant. Sorry. In a wave front for constructive interference. For constructive interference in what type of wave? Means actually the condition is even if the angle greater than phi c still we need constructive interference. See here we are talking about propagation of a single wave through an optical fiber. Which problem are you in? In the optical fiber problem or in interference problem? Actually I need to know the concept actually can you elaborate more the constructive. So, on the interference. Yeah. So, ignore this is the different thing it came in because I was doing critical angle at that state there was a question asked in the morning I did not have the figure with me. So, I brought it back ok. So, let me let me go back to your question. See the way one looks at it is. So, we all understand the electric field is a vector correct. Now, supposing I am adding several electric fields. Now, remember I am not adding intensity superposition principle tells me it is the vectors which have to be added. You have learnt vector addition in your algebra classes. So, how do you add two vectors? Let us suppose they have the same length first. Now, if they are same length then the maximum vector addition will be when head of one of the vectors is the at that point you start the tail of another vector at there at the same line correct. Then the total length will be twice that length. But suppose now one of the vectors makes an angle with the other vector all right which will happen if there is a phase difference between them. Now, what will happen is you will have to use what is called the vector addition formula correct. Now, so therefore, the superposition principle will tell you that if you are adding two waves let us forget about that polarization part first. Let us assume that polarization direction is the same. If the polarization direction is the same I do not have to worry about vector character of that thing. So, I write down E 1 let us say cosine k dot x minus omega t is one of them. The other one is E 1 let us let us say again E 1 cosine k x minus omega t plus phi x ok. Now, you add the two cosine functions because what is happening is this wave is changing with space and time the other one is also changing with space and time. But there is a phase difference between them. Now, when you add the two by the standard trigonometric formula what you find is the difference in the phase that comes into the picture. And if the phase difference happens to be equal to 0 which is the example that I gave you then the two vectors add up to give me modulus of E 1 plus modulus of E 2. Since I have taken E 1 equal to E 2 that if I square it I will get four times. But suppose there is a phase factor there and it does not quite add up that way. Then I will get still get an interference see constructive interference means that the vectors add up to amplify the effect ok. That is the result this is a the word constructive interference in Young's double slit is used at those points where the phase difference is 0 or multiples of n lambda. In which case what happens is at the tip of one vector along the same line you put in another vector so that the length multiplies. Destructive means the other one is going the other way so that you have less am I clear ok. So, come back here. So, is this clear what is the difference this theta is not the same as the theta that you talked about anything else yeah. So, you say that ok I understand the mathematics. So, this cos theta is theta is actually that whatever this exponential I inside that that is that theta. So, when you are saying that you have an incoherent light you are actually making that phase difference between phi 1 and phi 2 completely random. And then we know actually take the average over it. So, you don't get because it will average out to 0. So, but there is an another factor which this E 1 vector dot E 2 vector which actually gives is it just some magnitude E 1 and then that unit vector direction. This is the theta he was thinking it is. No, no I am not talking about that what I am saying is that the E 1 dot E 2 individually E 1 can be a function of time as well. No, no, no not at all what I have done is to say you say this is a linearly. No, what I am thinking of is that changing amplitude of the field. No, no, no, no, no, no, no the linear polarized vector linearly. Ah ok. Linearly polarized thing where the pre multiplication factor is a constant. Ok. Accepting that it is a vector because it has a direction. Yeah, yeah that I understand ok. So, what what we are saying yesterday we had a lot of discussion on the same thing we said look if E 1 and E 2 are vectors which are perpendicular to each other irrespective of what happens to that phase factor ok I will still get it 0. The phase factor is not important there and because of that two perpendicularly polarized light will never interfere. Am I have I made sense now ok let us come back to coherence because that is actually at the heart of whatever we are talking about and and that might help you in partly understanding what the question was. So, the point that I was trying to make is that the normal light sources that you see here nowadays of course there are many LED light sources I am not talking about that normal incandescent bulbs. Now they are what are known as thermal sources typically what you do is you heat up something and then of course that emits light you know I mean you it sort of becomes red hot then white hot and things like that. Now the when you say it emits light. So, basically it is emitting let us say an electromagnetic wave, but every 10 to the power minus 8 seconds it is emitting a different wave am I making sense because it is lifetime for excitation is only 10 to the power minus 8 seconds. So, it gets excited drops back release as a photon or a electromagnetic wave after 10 to the power minus 8 seconds it will be another. So, these two electromagnetic waves that it has released they are random waves there need not be any correlation between them right because the whole thing has started from the beginning again after 10 to the power minus 8 seconds. Now which means supposing I ask the question what is the distance which is the first wave which was emitted at let us say t equal to 0 has travelled when the second wave comes out. So, it is 10 to the power minus 8 which is the lifetime multiplied by the velocity of light which is c. So, therefore, which gives me a value of 3 meters correct because 3 into 10 to the power 8 into 10 to the power minus 8. Now which means that supposing I have a thermal source here it has emitted a photon. Now if I am observing it within a time or within a distance less than 3 meters then it is the part of the same wave does it make sense if the part of the same wave. Now since it is a part of the same wave if I look at points on that wave two different points on that wave I can always calculate if I know what is its wavelength I can always calculate what is the phase difference which means in such a case on an average over a 3 meter distance the wave has phase relationship it may or may not, but I am saying that is the maximum that can be because you could be looking at when one of the waves is ending the other one is starting, but on the other hand once you have exceeded a distance of 3 meters what is released is another wave and there cannot be there two different things. So, therefore, there is no relationship between that does it make sense. So, this tells me that the coherence length that I can talk about for a thermal source may be only of that duration. So, coherence is existence of a constant phase relationship between parts of the same wave or between two different waves because it is also possible that two different waves have a phase relationship because very often we divide the same source into two just what is done in Young's double slit experiment. We have one so slit from there we the waves come out and we let them fall on two different slits. So, therefore, we are dividing that way from well the point is this is this is to a to a certain extent the coherence is there, but way more than 3 meter I guess. No, no, no 3 meters is what I said is the maximum distance according to my calculation. This is obviously much more than 3 meter you can see it, but it is not a thermal source. Okay, you are talking about only thermal. I am saying a thermal source thermal source has a lifetime of 10 to 12. But the logic of getting 3 meter per second is the same whether you have a thermal or no. No, no, no, no. Because you have this transition of 10 to 12. It is a question of how much is the lifetime the laser principle is totally different. Okay, in case of a thermal source a random photons are emitted by the de-exciting atomic transitions that is all. See the when you heat it up okay a thermal source gets heated up. So, the electrons are raised to a higher level and on an average the typical lifetime is about 10 to power minus 8 and then there is spontaneous emission. Laser the whole idea is it is not spontaneous emission. This is clearly more much more than 3 meters. So, thermal source does not have that. Okay. So, therefore, if we want a stable interference pattern, you see the interference pattern that you observe we want them to be you know visible for some time at least duration of the experiment which might be couple of hours. Which means over whatever the duration is my sources must retain a constant phase relationship. Now, obviously I cannot assure that if I am talking about thermal sources but if I am dividing the same source even though different sources are coming out but it is always a part of the same source where the phase relationship is there. Am I making sense or not? Okay. So, that is why we always like to split the same beam into two when you do an experiment. This is the typical way in which experiments are done. There are other ways of doing it also. So, what we said is that in case of actual interference of two or more light waves the amplitudes and phases vary with time in a random fashion and the instantaneous flux naturally will also vary randomly and the time average will be 0. Now, let us look at since irradiance, irradiance or the intensity or whatever you like the we would like to define it more as a time average. This is very important to understand that we assume it without ever realizing it. When I am doing an interference experiment I am assuming interference pattern that I am seeing is a time average pattern. The point about this time averaging which I have defined here is that this quantity is taken to be stationary which means your what is your origin of time is not important. You take average over any long period the intensity the average that you get is the same and that is essential for getting a stationary interference pattern ok. Now, let us look at a typical coherence experiment. So, here what I have is this what I have done now is ignore that polarization. If I ignore the polarization I do not have E 1 dot E 2. I simply have E 1 into E 2 because both of them I have taken to be in the same direction. Now, so I have got by I is equal to I 1 plus I 2 plus 2 times real E 1 times E 2 star. Now, this is a typical interference experiment setup. You have a source S and then you have two slits one is S 1 and one is S 2. I have not even cared to write the picture in a symmetric fashion. And then of course, you have a point on a screen. A wave gets divided here goes one path goes here and then comes there another path comes here and goes there. The phase difference that arises between these two is due to the path difference that is there between the two and it is this path difference that you teach your students how to calculate in a young service. Then you say a path difference you know I if it is multiple of 2 pi then you have 1 lambda etcetera phase difference of. Now, suppose in going from S to P that time taken by path 1 is T and the path 2 let us say takes T plus 2 tau. We have T plus tau. Then the interference term is two times real part of a quantity which is the expectation or average value of E 1 at T, but E 2 at T plus tau. See what I said is supposing at T equal to 0 something was emitted at at T it reaches here, but this one is reaching at P at time T plus tau. So, what I am trying to do is this I want to find out a relationship between what happens to E 1 T with E 2 at T plus tau and this is our mutual coherence. And of course, I can talk about a self coherence by simply saying they are at the same time. No, no self see the point is that there are there is no no no it is not background you have one wave which is going by one path correct. Now, you see it is also possible that wave itself has lost phase relationship on the wave because it depends upon what type of wave you had. So, if a phase correlationship which is time independent phase correlationship exists between the different parts of the same wave this is the example that I gave you. For example, in case of a thermal source even I will not have a phase correlation after certain business certain time. Let me come to two phrases which I was asked that I must talk about. So, there is a thing called temporal coherence temporal meaning thereby time. Now, suppose I have a source which now remember we always say monochromatic and nothing like a truly monochromatic source. All sources that we have they may have predominantly say take even if you take your sodium light in the lab right. Now, you tell them that on my average wavelength is what is it 589.3 nanometer right is that correct or no yeah. So, but on the other hand you know that sodium occurs as a doublet actually 5890 and 593. Now, so therefore, there is a spread of 0.6 nanometer in the sodium doublet. So, you do not really get a truly monochromatic light. Now, if you do not get a truly monochromatic light suppose I have a source which emits wavelengths well I could say in the region, but let us say two wavelengths one is lambda another is lambda plus delta lambda. Now, I want to consider the difference after a time after a distance L at time t. Now, what we are saying is this that suppose look at this is this is the thing that I want to talk to that the phase difference between these two waves see I have emitted two waves one having a wavelength lambda another having a wavelength lambda plus delta lambda. So, since the wavelengths are different the corresponding wave vectors are also different. Now, as a result the phase difference between them when they travel a distance of L right should be K 1 L minus K 2 L and K 1 L minus K 2 L I know how much is that. So, this difference that I have got is what I am measuring. Now, I have to give a prescription of when do I consider the sources to be incorrect the if this delta phi which is K 1 minus K 2 times L that I have written down. If some you know I mean you have to give some prescription of when you consider the phase relations should be not important. If it becomes as much as one then we say that waves are not incorrect. Now, look at what it means it means that 2 pi by lambda which is K 1 minus 2 pi by lambda plus delta lambda times L equal to 1 and I assume that delta lambda is smaller than lambda. So, that my L is lambda square by 2 pi delta lambda and this L that I have written down there which corresponds to the situation when my phase coherence is lost that is what I call as the coherence length and the corresponding time which is obtained by dividing L with C ok. So, L you can rewrite L lambda square by 2 pi delta lambda as C by delta omega and so, this time tau is the corresponding coherence time. Temporal it means that the this happens to be just delta omega this is a very important relationship. You realize that well I could have called it delta tau and then you would have found it much more amusing. If I had called it delta tau you would have immediately realized that delta tau into delta omega is of the order of 1 ok. This is a very familiar relationship right. This is reminds you of uncertainty principle. Now, it turns out that we are not exactly talking about quantum metallic and uncertainty principle, but it seems like in optics there are equivalents of that. So, basically what it says is this that uncertainty in the measurement of frequency or energy because well frequency omega let us go back to quantum mechanics a little bit. Frequency omega h cross omega is your energy. Now, if there is an uncertainty in the measurement of that, then I know that there is one uncertainty principle related to time and frequency also ok and this relationship is almost identical, but sources are different. So, for instance to tell you what it is supposing I consider a wavelength spread about 0.1 nanometer and wavelength of 500 nanometer, which is typically what you do right 600 is what you talk about. Now, the coherence length then you remember the coherence length which you talked about was lambda square by delta lambda. So, lambda square 500 into 500 delta lambda is 0.1, which tells me the coherence length is about 5000 wavelengths which is typically 2 millimeter. What it tells me is this the interference visibility will be small for path differences which are bigger than this. So, coming back to the question that was asked see on a screen you do not see too many fringes and the reason is that from the two slits you take two paths to a point on the screen and calculate how much is the path difference. If that path difference is much bigger than 2 millimeter, then the interference will be essentially lost because by that time by temporal coherence principle that I talked about the there is no coherence between those two ways and that sort of limits the number of fringes that you can see on the screen. Supposing you are looking at white light now when you do it white light I sensitivity is maximum at about green 550 or so. And you know the sensitivity falls to almost 0 at the blue end or and at the red end. Now so this gives me from 400 nanometer to 700 nanometer with a maximum sensitivity at 550 gives me a spectral with about 150 nanometer and coherence length in such cases is about 3 to 4 wavelengths. This is the maximum number of fringes that you can see on either side. So, it is not just a question of coherence length, but even our eyes because we want to observe it with our eyes that also has a point there ok. This is a little more tricky I am talking about spatial coherence. Now basically we talked so now about the time coherence. Now suppose I look at the correlation between light at different points perpendicular or transverse to the direction of propagation. Now look at what I am talking about. See you have a source of light which is sending out light for your 2 slits. So, here I by 2 slits I am meaning a screen now not the screen in the usual observation sense. So, I have a screen there which has 2 slits there. Now this source is in a direction perpendicular to that slit. Now the question is that what is the relationship between the wires that arrived in a direction perpendicular to a direction of propagation ok and that is what spatial coherence is all about. So, just let us look at what it is. See let us suppose I have a normal lines source, real point source we do not have, but supposing I have a line source. Now if I have a line source of width delta and I am looking at it ok that let us say to a point p 1 and p 2 you can think of them as your slit position. Now the thing is this that let us look at supposing this distance is l and the width of the p 1 and p 2 distance is d. So, you look at it that from the center of the source let us suppose I take p 2 from. So, if I am looking at the left edge the left edge goes by along this path ok and from the center also there is a path that could go there. Now this difference that is between this and that you know what we do we drop a perpendicular here and find out what is the path difference and that is d by 2 sin theta. Now if you look at what happens to the right edge. So, the thing in the left edge the distance difference between the path from the middle of the source to the left edge is d by 2 sin theta which is there, but if you look at the right edge then it is d by 2 sin theta less. In other words from the two edges if you look at p 1 and p 2 I get a total difference of 4 pi d sin theta by lambda. See what I meant is this that at a particular p 1 or p 2 the difference is in phase is 2 pi d sin theta by lambda and so between the distance p 1 and p 2 it is this ok. So, this tells me the waves which are in phase at the point p 1 will be out of phase at the point p 2 by this amount is it does it make sense or not. See I have a long source and I have two slits there. I am trying to find out that what is the relationship between the waves that have arrived ok at the two slits. Now there is a phase difference between them because as I have shown you that if I had a line source the there is a path difference because p 1 and p 2 see had it been a point centrally located there is no path difference between the waves which arrive at p 1 and p 2. This is what will do in your Young's Double Satellite. You assume that a source is centrally located and is a point if it is so then the path from the source to p 1 is equal to the path from the source to p 2, but I do not have that. Supposing I had a line source then they will arrive with a phase difference between the rays that arrive from all parts of that source to p 1 and p 2 that is given by 4 point is sin theta by lambda. And it is this quantity again we say that look this tells me sin theta I can calculate what is sin theta ok that is delta by 2 L and so therefore, if I again say that if delta phi equal to 1 the coherence is lost that tells me that this distance d that between the slits cannot be more than this. Let us go back to 1917 the though he did not quite use that word, but the originator of the principle of laser has to be credited to Einstein. Einstein introduced the concept of what is called a stimulated emission the whole idea of. So, let me spend a bit of a time on what is stimulated emission and then we will spend a bit of a time on what is the principle of laser. So, what he said is this that if you want to describe the interaction between matter and radiation this is what he did he is interaction between matter and radiation then you see when there is radiation electrons in the lower level will absorb that radiation and go to a higher level. And the electrons in the higher level after some lifetime will again fall. So, one is an absorption of radiation the other one because it goes up there and comes by itself we call it spontaneous emission. But what Einstein did is this he said that we might be able to induce radiation. See spontaneous radiation is something which a system does by itself, but induce radiation means that if. Yeah. Actually the difficult part is obviously this course is supposed to be for the first year B.T.E.C. student, but the difficult part is that when you know in the back of your mind that you that there is really nothing as spontaneous emission. Yeah. Because everything is a stimulated emission even the spontaneous emission that we think is actually a stimulated emission. What sense? Because if you don't have any kind of perturbation it cannot just go down to the lower level you need to have a time dependent perturbation. So, firstly that there is something called a natural line width of every level. Yeah. A level is never indefinitely sharp. See the even if you believe if I believe what you are talking about see in principle there is a relationship between width of a level and the lifetime in that level right delta T delta nu is of the order of 1. So, therefore, if one has a very sharp level then the time that a particle would spend there would be essentially infinite this statement is correct. But nevertheless what we have in reality are levels which are closely packed and they have certain amount of width. And so, therefore, every atomic level has certain lifetime. See if you believe only in Bohr's model there is nothing like a lifetime. Einstein's theory of this period which is known as Einstein's theory of semi classical theory of radiation. So, it is not quite quantum let us let us leave quantum mechanics for you know different days. I mean I am not that I disagree with you I am just saying that we are not discussing quantum mechanics here ok. So, what we are trying to say is this that what Einstein suggested is that in addition to spontaneous emission that takes place whether you do anything or not. It should be possible to induce emission by having merely by having radiation present in the system photons present in the system and this is he. So, let us look at the situation here what we are doing is this that look at these are the typical energy levels we have. But you know that even in Bohr's theory there is a postulate made that the electron from the higher states makes a transition. In fact, Bohr's theory postulates have nothing to do with perturbation theory, but these are these are semi classical in him ok. So, these are my energy levels E 1 E 2 E 3 etcetera and I know that whenever there is a transition from one level to another a photon is there. Boltzmann had provided a prescription of how many particles will be there supposing I have a closed system let us suppose I have got 2 or 3 energy levels. I know that my electrons must occupy different energy levels and he says that at thermal equilibrium at a temperature T the distribution the ratio of N 2 to N 1 depends upon E to the power minus E 2 minus E 1 over k T. In other words the population of the level exponentially decrease as the energy goes up at a given temperature. So, as a result by nature the number of electrons which will be there in higher and higher level will be much less than those which will be in lower and lower. Now, so this is this is Boltzmann factor. Now, when we talk about it these are Einstein's notation spontaneous transition probability even though I have shown here 3 levels let us talk just about 2 levels N 2 and N 1. I have an energy E 1 in which N 1 number of particles or electrons are there energy E 2 where I have N 2 number. So, according to Boltzmann N 2 by N 1 is E to the power minus E 2 by k T by E to the power minus E 1 by k T and N 2 is much less than or less than N 1. Now, if there are particles in E 2 they will go to move to the level 1 by what we call as a spontaneous. It is spontaneous because it is in the nature of things that electrons which are in the higher level will come down to the lower level. So, that is in that context only I am using the spontaneous. So, therefore if there are N 2 number of particles in the level E 2 then since the transition probability incidentally A 2 1 is called an Einstein A coefficient is simply N 2 into A 2 1 because A 2 1 is the transition probability. The mathematics is absolutely trivial. Now, after that let us say that there is some radius and present. Now, this radiation frequency is the frequency associated with transition from E 2 to E 1. In other words I define nu is equal to E 2 minus E 1 by h. The question is that the transition probability from 2 to 1 which is being induced it happens only if there is a agency trying to help it. Now, this is called B 2 1 and not only that that in the presence of these photons there would even be an upper upward probability of transition going from 1 to 2 which is nothing, but if there is radiations present in the medium and there is an energy level E 1 it is going to absorb some of them and go to it that is a absorption. So, I have got I have defined three things. One is spontaneous emission which does not depend upon any radiation present there, another is induced emission which takes place when the radiations of frequency which is E 2 minus E 1 by h is present in the system ok. Now, I want an equilibrium now what happens at equilibrium? The equilibrium means by definition the number of transitions into the upward level must be equal to that in the downward. But you see downward I had two processes, downward my processes where one process is N 2 into A 2 1, another one is again N 2 into B 2 1 into U nu this is the induced one and this is the absorption. Now, you equate them and you get an expression for the energy density of the radiation that is present in terms of A B and this absolutely trivial arithmetic and you can do it easily. Now, what Einstein tried to do is to say this is the energy density of radiation. So, the next formula must be valid. So, I said how can I make this formula that I have arrived to the plans formula. So, is his postulate was make B 1 2 equal to B 2 1. In other words the induced transition probability, induced emission probability is equal to the absorption probability. And you say then also you need A 2 1 by B 2 1 is this factor which must be familiar to you in Planck's law. So, if you do that then what I get is in thermal equilibrium this relationship is valid. Now, let us look at this relationship that this tells me that for ordinary optical sources if I take typical temperature is about 1000 degrees ok. And I am supposing in visible region so that I know how much is new then what you find is the spontaneous emission process dominates and the induced emission is almost negative. Now, however what happens in a laser is radius and density certain not all modes certain preferred modes build up to a value which is such that the induced transitions become dominant. And this is the principle of laser. So, let us look at what I am trying to talk about. I am saying that look the. Sir. Yeah. What would happen if the thermal equilibrium condition is violated? Well equilibrium by definition means the number going up is number coming down otherwise you do not have an equilibrium. No, in some of the books I read yeah it says that the still that ancient relation holds good even if the thermal equilibrium condition is violated that I do not understand what is being said is that we have said that the I do not know which book and what it is talking about, but what I am trying to say is that by definition my equilibrium is one where a steady state has been reached which means number of transitions that is going up that must be equal to the number that is coming down that is my definition. This is the thermal. The question of particles that excited state becomes more than the part of the traditional part. No, I have not yet I am that is my job now ok, but the thing is this. Now let us look at it that is it possible that my spontaneous emission will be less than induced emission under normal circumstances no and that is because of this curve. You say this is the Boltzmann curve n 0 e to the power minus e by k t where the you can see the population at level e 1 is always more than the population at level e 2 this is Boltzmann distribution. Now, however if I could somehow make the energy levels like this there is the distribution still, but supposing e 2's population is here this is population and e 1 is here. In other words if I could somehow rather make the population of the higher level lower than the population of the lower level which is not a normal Boltzmann equilibrium condition. Then by the formula that I told you it should be possible to generate that induced emission would be greater than the spontaneous emission. How does this happen is how does not do it? This process is known as population inversion ok. The typical laser has essentially two components. There is a component which is called an active medium where which is also called a gain medium sometimes because what happens there is I get multiplication in the number of photons. Now that active medium could be in any state solid state liquid state gaseous state could be collection of molecules atoms or whatever, but there are certain mediums in which these work all right. Now, there are other situations there that induced emission happens to be in the same direction as the primary beam and is highly coherent etcetera these all you know properties of laser, but typically this is what is happening. I have a gain medium and in this gain medium what is happening is that the intensity I that is going in is getting amplified by an exponential factor e to the power g m. I have still not said how population inversion is achieved as yet, but let us suppose there is a mechanism by which it is achieved. Let us look at one possibility. See what happens is left to itself the population of the lower level will never be lower than the population of the upper level, but I can do something else. Supposing I work on a I cannot have a two level system then because even if I somehow or other persuade the electrons by pumping energy to go to the upper level they will immediately fall down to the lower level. So, that is not going to happen. The minimum for laser action is to have a three level laser. So, suppose I have a lower level and I have a level which is much higher in energy, but in between the two there is another level which I have called as you have energy e to. Now, notice that what I do is I can persuade the electrons from here directly to go to N 3 by pumping and just give it power. Now, when the electrons make transition from that level they want to come down they can either come directly or they can come by an indirect process. They can first park themselves in N 2 and then from there to here they can come. And then if I have this right that pumping rate once it has reached certain threshold depending upon the relative life time between this level and that level then it is possible that more and more particles will be parked here than they were here because you remember see what we are doing is this. We are pumping electrons from the lower level. So, electrons are decreasing from there we are taking it to the upper level, but the upper level electrons are not directly jumping back. So, part of that is parking at an intermediate level and then they will ultimately come. So, there is a threshold beyond which the population of the level 2 will become more than the population of the level 1. So, there is a population inversion achieved there right. This is incidentally not a very you know efficient process and the power required for this is very high. So, if you want to do pumping what people do is actually more a 4 level process. The 4 level lasers are much more practical than a 3 level laser. See here what happens is again you still pump to the that 4th level. There are 2 levels in between there is a transition probability T 4 3 T 3 2 T 2 1 and supposing this transition probability and this transition probability that is these are much smaller than this transition probability. Then what will happen is even at moderate power the population inversion may achieve may be achieved between the level 2 and level 3. Can the pumping be just directly to from E 1 to E 3? E 1 to E 3 transition. Why should we unnecessarily take to E 4 and then. You can certainly do that. But it is see the point is this that usually what happens is these require a little more difficult processes because they are never direct transition. Direct transition. Say yeah. What do you mean by more than half of the population must be there in the higher energy level and the second is why they are why they have been pumped to to the lower level only why they jump to the lower level only why not directly to the even level. No, no, no. From E 3 to E 2 they jump. It depends you see between any 2 levels there is a transition probability that exists ok. It is not that the you know you have just given level and they will themselves come there. There are these they depend upon say remember that they are not always direct levels. If you have seen any of these band diagrams you will realize that they are not at the same level sometimes one of the processes requires a conservation of momentum which is lot takes a little more effort than this. So, therefore, this there are indirect transition and direct transition. So, when you pump it to the fourth level it will automatically come down to the first level if you do not do anything. But there are also possibilities that they are doing this, but what happens is some of these level have a higher life time than the other levels. So, therefore, they will be parked there little longer. So, as a result with time gradually there will be because from here you are continuously removing electrons. There things are coming in and there is a little more life time. So, they are resting there. I mean bad examples, but imagine that the people are basically being removed from a place by let us say train which keeps on coming you know I mean ok. Let me give you the typical example of a Bombay local trains ok. The morning if you are in some place like let us say church gate or Burrivili you will find that there are thousands and thousands of people and they are being continuously removed. Now, they are being continuously removed, but of course the people are not dropping back from the same place, but people are coming in also. Now, if that is the only process that happened then the population in the lower level you cannot or population of Burrivili you cannot reduce in the morning. But supposing there is a intermediate station where just you know there is a stoppage for a little longer time. Then what happens is that some of the population are parked there and as a result the population inversion occurs between that level and the ground level or in the this example the level N3 and N2. I would like to know which are those levels where they are being put down and why. Ok. So, they in order that in order that that is to be answered the levels that are there would depend upon which material you are ok. And the lifetime of that levels are known you see I am not talking for example, if you take ruby there is a particular type of levels. No, no, no. The lifetime is more at the lowest level because that is where it stays ok. So, in the case of. Lifetime is very small at very high level, but intermediate level I am talking it is a metastable. Metastable means its lifetime is longer than the higher level which is understandable again. But from the lower level you are continuously pumping that is why they are going away. You are removing things physically by supplying power ok. So, lifetime see there are suppose there are just three levels lowest level is ground state. Ground states has a very long lifetime. For example, in this four level system. Let us say four level. I am interested in four level. Let us say in this case you mean to say lifetime is maximum in N1 and least in N4 and decreases from N42. They are not decreased like that, but because no. In that case N2 becomes N2 has much higher lifetime than N3. This lifetime it is a ground state it is basically infinite lifetime. That is true. This is a very short lifetime. These two are levels which have not this short lifetime or this long lifetime. They are called metastable states. I agree about that. I wanted to know actually the concept of metastable state how does it arrive like like N2 and N3. N3 should be metastable higher than N2. On what basis the level of energy decides lifetime? It is a property of the system. It is a natural phenomena of the system. What can find out how much is the lifetime by doing an experiment or things like that ok. Sir excuse me sir. Sir in the case of ruby laser it follows three level pumping scheme. In the case of HG and laser it follows four level pumping scheme. So, as far as three level is concerned what happens in there is three level is there ruby is nothing but the AL2O3 plus ER plus 3IL. I mean I don't know about those details so why don't you explain to everybody. Yeah so chromium ions are basically active centers. And whenever we are pumping them optically so what happens blue and green components are absorbed nearly 5500 angstrom is absorbed and ultimately we are getting red laser beam that is 6,943 angstrom we are achieving ok. So now in that case what happens is that ruby laser the CR plus 3 ions will absorb that ok. And it goes to the higher level. It radiates some non-radiation ok. It gives up in terms of heat and it achieves the metastable state. And whenever nearly 50 or 55 percent of it accumulates in that particular metastable state spontaneous out of 10 to 6 you can say out of one million one or two or maximum three chromium atoms can spontaneously de-excite themselves. And that particular photons, red photons will stimulate other metastable chromium atoms. And within few milliseconds only we can have this sudden avalanche effect. Ok. And LOS here will be there. Alright, thank you. I am not an expert on laser but yeah. Sir the question is this metastable states are permanent energy levels of the material. Sorry. They are the permanent energy levels of the material that specific material or they are something else because. They are not artificially constructed levels. They are levels which exist in the atomic spectra of this system ok. And the thing is that the lifetime of some of those levels like he made a statement these are non-radiative transition. But basically what happens is you notice that they are not put in the same line which means in order to come from one to another you must also satisfy a wave vector conservation ok. So, something is coming in there to the third one but because it has a little longer lifetime it waits there a little longer. You are pumping at a much higher rate. Now when you are pumping the spontaneous emission is taking place but because of the indirect transition some the population in the intermediate level is building up. And like he gave a prescription that once it becomes more than 50 percent the laser action can take place. Now you see the after that we have we have come to the end of it. Sir one question I have. Yeah. Sir in that four level system or any six level system, five level system random energy levels are metastable states. Ok. So why and how do you. Two energy levels are metastable states. Random it is not ordered like fourth one only is there or second is metath third is that ok. You are absolutely right lifetime of hello sir hello. Is more determined by what is its energy level ok. They it is true see the only population of the energy levels determined by the Boltzmann relation. So lifetime. But what is the lifetime depends upon many other things. Many other. What kind of things. Sir can I comment on it. Yeah please go ahead. The lifetime of a of a particular energy level is decided by selection rules for a particular material. Those selection rules are defined in Kaplan's book and you can go through and see them. Clarify your doubts. Ok. Thank you.