 Good afternoon, as we start the last my last lecture in the series, I will complete the optics section I was doing yesterday, but before I do that as usual I will take up some of the problems questions which came up in the last as a questions during the last lecture it turns out that even though I was talking about optics most of the questions which came yesterday were from superconductivity and of course I will not be able to spend that much of time on superconductivity. So what I will do is all those superconductivity questions will be answered on the forum and I will only take up today the questions related to only optics that is the portion which I did yesterday. So let me let me pick up the questions which came yesterday. The first question I pick up is from 1, 2, 6, 3 Techno Salt Lake, Kolkata. The question is came from a statement that I made yesterday that optics people like to differentiate between irradiance and intensity. So the question was what is the difference between irradiance and intensity? It is a matter of actually definition the way the optics community uses the word as I told you yesterday that I define the pointing vector as you know E cross H and the average pointing vector is because it is the average of the sin square of the cosine square term is E 0 cross H cross. Now this direction of the pointing vector is along the direction of propagation. So this then becomes the magnitude of the pointing vector is along the direction of k. So I times k I think yesterday I wrote k by k that is not correct. The irradiance is basically the magnitude of the pointing vector namely half E 0 H 0. Now if you look at 1 to write in down in terms of the electric field alone then it happens to be n by 2 z, z if you remember is the square root of mu over epsilon 0 which is the impedance of the free space. So this times E 0 square and since the interpretation of the pointing vector is the amount of energy falling per unit area per unit time. So this is roughly the irradiance that the magnitude of that the pointing vector talks about direction of flow but irradiance is just the magnitude. So what is intensity? Intensity is very closely related thing accepting that what the optics people like to do is to talk about the supposing you have a source and talk about a solid angle around that source and find out that supposing I take a unit solid angle, I take a unit solid angle and look at an unit area there in. Remember that solid angle does not define how much area you are inclusion because it depends upon what is the total distance from the source I am assuming a point source. So therefore this intensity which the optics people sometimes indicate like this is power per unit solid angle per unit area. So basically if I take a point source and if the radiation is perpendicular to the surface then I know that the solid angle around it is 4 pi and hence what I have is this is this quantity is nothing but I divided by 4 pi. So that would be my I. So it is a matter of definition nothing great in it and as I said very often we use these interchangeably. The second question says that we know how to this question came from 1305 Kavikul guru institute of technology at Ramtech. The question was that how do you analyze a circularly polarized light? We all know how to check for a linearly polarized light because I take another linear polarizer and if I put rotate around it then of course there will be a position at which no light will come out which is why it is the second polarizer is sometimes called an analyzer. So the question is that how do I determine whether the polarizer given to me has had this either circular or elliptical. The point is that you do it exactly the way you made it. If you recall the way we made a circularly polarized light I mean polarizer is to say that I take two mutually perpendicular direction and have it at same amplitude. Usually the same amplitude is simply obtained by starting with a linear polarizer whose axis is aligned at 45, start with unpolarized light and have a polarizer whose axis is aligned at 45 degrees to the x axis let us say. So that the component of x and y for this linear polarizer is the same and after that you pass it through a quarter wave plate which has the ability because it has two different refractive index indices for the two directions. So you pass it and measure it in such a way that it introduces exactly a phase difference of pi by 2 and if you do that then of course you will get a circularly polarized light. Now you could do exactly the same thing in this case because you see if you just had linear polarization or unpolarized light a calcite crystal with a quarter wave plate made out of a calcite crystal will not have any effect on a linearly polarized light or unpolarized light. On the other hand if I have to pass through a circularly polarized light is to pass through it then of course it will be effected. There are many interesting ways people have suggested in order to check for circularly polarized light one of them is that hold the polarizer close to your eyes and look at the reflection of your eyes on a mirror and you will find that if you rotate it you might find no light and that is because that the reflected light and the incident light at the surface must satisfy the electric field due to incident and the reflected light must be equal and opposite and because of the phase differences that is there it turns out to be dark but that is just a test. The there is one question which I will take up a little more in detail but there is a question or rather an inquiry from 1158 NIT Durgapur. So NIT Durgapur has pointed out that this is known as the year of light as you know that the United Nations had declared 2015 as a year of light and he says that he would like to know what sort of researchers are going on in light in this country. Well obviously I will not be in a position to tell you all about it in next 2 minutes but we have had many major centers of research in optics. Some of them are for example Tata Institute of Fundamental Research Bombay, Indian Institute of Science Bangalore, the Center for Advanced Technology at Indore and lots of other IITs and things like that. In fact the there would be in case you people can find out shortly an issue of special issue of physics news which is the Journal of the Indian Physics Association which will have articles only on light and celebrating the year of light but yes a lot of research goes on in that. The last question which I take up is from Biswakarma Institute of Technology Pune. The question is that they want little more clarification on what is evanescent wave, what is its use? I talked about it when somebody asked a question but let me repeat it again. See what we said is supposing you have a medium does not matter what medium I take because it is not necessary that evanescent wave have to be talked into in connection only with optical fiber but let us suppose I have a refractive index n which is greater than the outside refractive index the sorry the this is I am having a reflection the refraction from a medium with a you know I mean within this n so that suppose I have this incident at an angle theta and such that this angle theta is greater than sin inverse of 1 over n 1 over n because that is the refractive index of the other medium then what we showed is that this is totally reflected. Now it is totally reflected but if you looked at all our formulae that we derived we said that supposing you take this plane the surface plane to be the xz plane then if you look at the electric field just outside the surface the it does not oscillate but what happens is that I get a profile of this type we will get a profile which is like this. You have seen this picture in damped harmonic wave and it sort of goes as e to the power minus alpha y, y is the distance from the surface this is known as an evanescent wave. So, in other words it says though the total energy is completely reflected into the medium in which the reflection took place the in the other medium in this case here in my example the certain amount of field leaks through and that field as you go away from the surface it quickly exponentially decays away and typically the field is can be found and measured within about 2 or 3 wavelengths of light and so therefore the this is known as an evanescent wave that is the wave which exists in the other medium even when the total internal reflection took place. Now yesterday in that question session I pointed out that you know people ask what good it is there are many uses of that but in connection with the optical communication which we are talking about the point is that suppose I put a you know I mean let us suppose this is an optical fiber remember the way the optical fiber works I have a core here and I have a cladding here and I of course have total internal reflections here and then in this region I have an evanescent wave. Now so therefore if I can bring a sensor very close to it I can pick up that wave now supposing the original light wave is actually carrying some information in the nature of there is a time variation so that time variation is still there because what is important is the space variation is goes as e to the power minus alpha y but if the original ray of light or the wave had some variation with respect to time the same profile with respect to time is even in this case there. So in other words if any information was coded in your original transmission that same information happens to be there even when I pick it up from the evanescent wave and that is why this has quite a bit of a use as a coupler in fiber optic transmission because when a fiber optic is going around from let us say one city to another city and you want the same information be put up picked up you can tap it by using a coupler ok. With that we complete the questions that came up there and I will now go over to the slides for today. So today my primary interest will be to look at light as for communication and yesterday we had just about started talking about what is coherence when we ran out of time but I would like you to be reminded and in fact it is important that you all understand little more clearly than is written in the book of the idea of a coherence and why it is important because you cannot have interference of light if the beams are not coherent. In fact that is primarily the reason why we pick up in double slit experiment the divide the same source so that I have a division of the you know the same wave front I pick up the light source. So the coherence is a fixed phase relationship between the you know phase up wave in a radiation of single frequencies let me explain this with a little clarity. Now suppose I am talking about two beams of light then if I can somehow or other arrange such that that as time proceeds then the phase difference that is there between the two waves remains constant of course you would say that this cannot happen well it may not happen for all types of waves but on the other hand as I told you that we can arrange it such that the phase relationship is independent of time. Now let me give you an example of what happens supposing you take a thermal source. Now thermal source is where the light is emitted by essentially by heating up a thing. So what happens is these in these thermal sources you excite an electron and as the electron falls back to the ground state it emits a light but but you see the problem of thermal sources that the typical lifetime of the electron in the excited state is about 10 to the power minus 8 seconds or so. Now so what that lifetime means is that if you are taking a thermal source that thermal source is giving different light and different waves in every 10 to the power minus 8 seconds. So in other words that if you wait for 10 to the power minus 8 seconds you will be you are likely to pick up another wave and if you pick up another wave these two waves are not phase correlated and so since I said 10 to the power minus 8 seconds it is the typical lifetime of a thermal source. What it means is if you multiply it by the velocity of light in vacuum which is 3 into 10 to the power 8 it tells me that the distance over which the phase correlation remains is just about 3 meters. So if you are looking at a wave which on the face of it looks like a single wave but actually speaking you are having you know one wave after another. So if you have waited for a distance more than 3 meters you are not going to get a phase correlation. And as I as we may they comment in the beginning this stable interference patterns are formed only when the radiation emitted by coherent sources usually we split a single beam into two or more beams as we do in Young's double slit or grating edge. Now in case of actual experiment of two or more light the amplitude and phase they can vary in a random fashion and the instantaneous flux at a given point will fluctuate randomly. And as we have said that irradiance is defined as a time average. So therefore if I am looking at a situation like this remember that this was the intensity expression and I told you that intensity irradiance expression and we said that this is what it is. Now what we are interested in it is in a time average because and the reason is that why I am looking for a time averages that the speed with which light is moving is very fast of course the fastest that we know. And supposing I am doing it with visible light I am depending on the persistence of my eye to look at it or even electronic instrument. So therefore the impression that is left there is for a fixed time. And so as a result what you will see is an average thing and hence I am interested in the average procedure. So this is what we have the we take suppose E1 and E2 arise from a common source then what we do there is that I have a source here it goes by path you know I am normally in a Young's double slit experiment you arrange such that this path and that path are the same to the two slits you know you do not have to do it you have a path 1 and you have path 2 it goes to slit S1 and you are looking at a interference pattern at a point p. So this thing this one this one has gone through this path path 1 this is the distance it has traveled path 2 is this. Now if I have I am to look at such a situation then the time taken by the path 1 let us say looks a little bigger. So it is a slightly more time than the time taken by path 2 because the distance is a little longer. One defines what is known as a mutual coherence function by the time average value of E1 t and E2 t plus tau because it is that if I am observing at a particular time what I am looking for is something which came from the source if it is on this thing you know if this took a time t this has taken a time t plus tau. So I am looking for the correlation between an electric field in one of the paths at time t in the other path at time t plus tau and this is and if you make supposing it is the same path that is I went directly from S1 S2p then I can define the same thing accepting that I have to replace 1 with 2. So that I find out whether there is a self coherence or not. Now so I said that suppose I have a source which emits normally though I would like to have a totally completely monochromatic wave. A monochromatic wave is one which gives you precisely a single wavelength but of course most of the waves that we have they have certain amount of width and suppose I have a source which emits wavelength at lambda and lambda plus delta lambda then the if I look at the phase difference after some after each one after I have traveled some distance l at a time t. Now then the phase difference between the two waves the my original source had let us say component two waves this is nothing but k1 minus k2l this is the phase difference k1 and k2 are the wave numbers and so therefore the difference is 2 pi by lambda minus 2 pi by lambda plus delta. Now I have to sort of decide or give you a criterion of when can I say that the waves are not considered coherent and well the optics people have picked up an arbitrary thing we say that look if the phase difference is greater than 1 radian then do not the waves are unlikely to be coherent. So therefore this is the thing and so this thing l is lambda square by 2 pi delta lambda that is your the coherence length that is the length over which the a particular source remains coherent right and correspondingly if that is your coherence length lambda square by 2 pi delta lambda very often people take it as just lambda square by delta lambda it does not matter because these are all estimates and if you look at the time then what you need is to divide it by so this is your l and you can write it in terms of the frequency or the angular frequency if you like by writing multiplying and dividing by this thing you know I mean if you had nu by d nu and differentiated it you will get lambda square by delta lambda. So that turns out to be c by delta omega which is 2 pi difference from delta nu and therefore your the coherence time which is simply l divided by c is given by 1 over delta nu or 1 over delta omega if you want to include that factor of 2 pi. So this is called coherence time that is the time over which a source can be considered to have maintained a phase relationship I have taken an example as an example consider supposing you are taking a 500 nanometer wavelength which is which is in the very visible spectrum and take for example sodium light which is 5 8 9 3 and minimum average but it has a spread of about 6 nanometer 5 8 9 0 to 5 8 9 6 but for calculational ease take a wavelength of about 0.1 nanometer width in a wavelength of 500 nanometer. Now so how much is the coherence length? So we approximately is this 500 nanometer square divided by this 0.1 nanometer which is I have ignored that factor of 2 pi. So this is about 5000 wavelengths which is nothing but 2 millimeter. Now notice what it says 2 millimeter is not all that bigger distance the. So it says that if you are taking a wave like that which has a wavelength of 500 nanometer but has a spread of 0.1 nanometer and if you are looking at the interference pattern at a such that their path difference is more than 2 millimeter then the visibility of interference prejudice will be small for such path difference. And that is actually the reason why on a particular screen when you are observing Young's double slit experiment though your textbooks will tell you that there is n equal to 1 2 3 4 God knows how much you can do they will talk about what is the 10th order spectrum 20th order fringe but actually what you see is never more than 3 or 4 fringes. And primarily because by that time you go to that distance the coherence between the two beams have been lost. So supposing you are doing a white light interference white light has a maximum sensitivity I sensitivity at the middle remember that white light consists of red to blue that or blue to red 400 nanometer to 700 nanometer thing and with a with a maximum I intensity I sensitivity at about 550 nanometer. So the spectral width which is this is the central wavelength. So the spectral width on either side is 150 nanometer and if you calculate using that formula that lambda square by delta lambda you would notice that coherence length is just about 3 to 4 wavelength and this is the maximum number of fringes that we can see on both sides in an interference pattern. And this is what makes the idea of coherence extremely important. Since I will be doing something more today I will be sort of glossing over this part but let me also talk about a coherence which is known as the spatial coherence. Now the spatial coherence is something like this that supposing I have a extended source. Now if I have an extended source it is not a point source then supposing I have this point P1 and P2. Now look at the path difference that one has to have if you want to for example if you look at the point P2. Now there is a path difference of the wave that was emitted from the left hand side of the you know this extended source over the wave that was emitted from its center and this is this is more than this ok. On the other hand if you sort of went to the right hand side then of course this would be less and things like that. So ray from the left edge to P2 travels an extra distance because it is travelling much more ok. And this from this point it travels a shorter distance because every time I am comparing with the path that it is compared with the center. And if you look at the path difference between the two this corresponds to a phase difference of 2 pi d sin theta over lambda where this is your d. And if you do that then we convert this into phase and so waves which were in phase at the point P1 will be out of phase by the time you go to P2 because there are differences there and you can calculate if you put in again the same delta phi equal to 1 you will find out that there is a distance after which the phase coherence is lost. And this distance depends of course on the you know I mean your length of the distance here delta which is the size of the source and wavelength and there is some constant factor which came from this 2 pi and things like that. So this is called spatial coherence ok. Having talked about coherence so let me let me before I quit this let me summarize that I we are interested in the coherence property of light that is because the interference the superposition principle will give me something different only if I talk about adding light which have you know coherence between them. So coherence is first thing that I understand must understand about light you of course realize immediately very glibly we say that I have a monochromatic source a strictly monochromatic source cannot have any it never goes out of phase because a strictly monochromatic source it means it only is giving a single wavelength we will we will come back to it when I have to talk more about it soon. Let me now give you a slight bit of overview about the light role of light in communication. So far as physics is concerned we are interested in 2 of those things which one of them is ok first let me sort of point out that in any communication your important points are 3 of them. First thing is you need a source you need a source to you know when generate a signal. So that is number 1 having done that the you need something to carry your signal I will I will sort of tell you what those are and finally when this signal reaches the other end you want to reconvert and detect this signal that has come. So I have a transmitter I have a medium which transmits it which is usually called in electrical language transmission line and at the end I have a detector. So these are the major component there might be many other things amplifiers etcetera etcetera but let us not worry about it. Now first major advance in the field of communication happened in 1876 Alexander Grunbel who as you know invented telephone. Now what he did is to start with because you he was interested in the voice communication to be taken from one place to another. So the way he did it is he had a method of converting the audio signal to electrical pulses at the transmitter end. So his transmitter is basically an audio to electrical converter having got the so audio signal is converted into electrical pulses having converted into electrical pulses he transmits it usually over cables and then at the end when you receive that telephone your instrument has the capability of converting the electrical signal back to the audio signal and that is how even today that is the basic principle by which of course the technology behind Alexander Grunbel's telephone has undergone a lot of change but that is still the basic principle by which we hear a telephone conversation. Now in order to do look at it let us look at the electromagnetic spectrum. So here the electromagnetic spectrum is shown some scale the wavelength is shown here this is 10 meter this is in decreasing wavelength I have shown 10 meter 1 meter then 10 centimeter reducing at equal distances by about one tenth 10 centimeter 1 centimeter 1 millimeter 0.1 then micrometer 1 micrometer 0.1 micrometer etcetera. The corresponding frequency which is in hertz the corresponding to 10 meters it is 30 megahertz 300 megahertz then the name changes it is 3 gigahertz or gigahertz whatever you feel like 30 gigahertz 300 gigahertz and again I have increased it by another 10 to the power 3 so I come to 3 terahertz 30 terahertz 300 terahertz and when I go there it is called petahertz which is 10 to the power 15 there are names even that side but let us not worry about it. Now if my signal if I am when I did communication with in this range the carrier was a coaxial cable. Now now comes some development or where the people were able to increase the frequency we will talk about why we need to increase the frequency and when you want to gigahertz that is your microwave communication this is what came to be established as standard during the second world war and you found that the waveguides were the solutions for that. And today we are talking about a much higher frequency range and it is in the terahertz or petahertz region and that is where the optical fiber communication becomes important. So the point is this that with the increase in demand for communication our requirement for now see the point actually is this why do we need to increase one frequency after another that is the question why cannot we still stick to this. Now the answer to that is the following say look at the voice communication in the today or in the old days. Now in order to transmit your voice supposing I am one person who is talking about and you know the I have to convert the audio signal to electrical signal and then transmit it. Now suppose I needed about 4 kilohertz of bandwidth for the purpose of one communication. Now there would be this actually is not good enough because there are other electronics etcetera will be involved and typically this will be let us say about 64 kilohertz after taking care of all losses and things like that. Now suppose at a time 10 people want to do the communication now then I need a width of 640 otherwise one person's voice will be superposed over the other person's. So if you want to talk about simultaneous communication which we are talking about we have to send them by different channels because I cannot superpose my voice over another person's voice. The other possibility is of course from the telephone exchange everybody has one separate line now that is again not a possible idea because that is an extremely expensive thing. So one possibility is that I have a different bandwidth allotted for the different persons now which requires that I need supposing I have n number of people 100 people I need 6400 kilohertz. Now so therefore when I said that we come to the megahertz region that is the coaxial cable type of communication I had a certain amount of leeway because my range was still reasonable and you realize I have already come to megahertz with this 646400 kilohertz but on the other hand the total amount of data that can be simultaneously sent over a channel like this is still limited. Now then came the growth during the second world war where as I said we went over to screen so then what we said is that I am here. So I am in the gigahertz region and this is of course you know being 10 to the power 6 and things like that I have a much bigger megahertz is 10 to the power 6 this is 10 to the power 9 so I can allot many more channels there. So I have a wider bandwidth because I can support more channels. Now our search for communication and the need for sending data, communicating data increases day by day. So after the microwave communications saw its limit has been reached then people were looking for that can we do better. At that stage the electrical engineers looked at the physics laboratory I mean because the question always is we are discussing physics why are we talking this. In physics people has been working with optics using lenses, prisms etcetera for years much before the even the subject like electrical engineering started and this light has a much higher frequency. So with the increase in the demand for communication the requirement of the higher bandwidth came and increase in the bandwidth requires the increase in the communication frequency. Now this is actually an empirical situation. The empirical situation that is there is that the bandwidth the as you know what is an empirical relationship an empirical relationship is something which has been found to be true I mean. So what is found is that the bandwidth requirement obviously bandwidth requirement is proportional to the number of simultaneous users. Now there is an empirical relationship which says that the bandwidth is equal to the central frequency divided by the quality factor of your transmitter. Now it turns out that the quality factor is roughly independent of frequency making the bandwidth proportional to the central frequency itself. So it is because of that that if I want a bigger bandwidth I need a bigger central frequency. Now if I want reliable communication I need two things. First is I must have a signal to large signal to noise ratio is called SNR by electrical engineers. Signal to noise ratio simply tells you the ratio of the signal power which you have fed to the system divided by the power that is lost due to noise. The noise could be a various variety strain noises white noises and all that. Now so one of them we already talked about that I must increase the bandwidth. The other is to increase the strength of the signal because say noise is not strictly in our control. We can do something to reduce noise but can we increase the power of the signal. So we said that whatever microwave communication see the microwave communication reached its limit due to a electronic restriction. What happened is soon its wavelength become comparable to the size of the separation between electronic component. And as I said that if when the communication scientist looked at the physics people they found a an window of opportunity. Because physics people were dealing with much higher bandwidths bandwidths which were which could be increased by factors of 1000 to 10000 in you know I mean going from the microwave to optical region. So the question is this that I want a communication remember I told you there are three things there I need a source I need a carrier and I need a receiver. So first question is are there light sources which could carry information. You would say what is what stupid question is this. Because light sources are there everywhere sun is a source of light moon is a source of light in some sense. But we have lights everywhere incandescent bulb tube lights so many things. So we have sources but the question is can that light carry information. Now remember light itself is not information. You do not want to send light from one place to another just for the heck of it. You want to code some information in it. Now how would I code information ok we come to that. Second question is that electrical signals were carried by either coaxial cable or waveguides is there a medium through which light could be transmitted over long distance. Now look at the second question you would say what is the problem. I mean I know that if there is light light after all comes from great distance from sun to coming to this place. Now the problem is that I cannot have a sun on this place just to carry sun is a very bright source of light. So the type of sources that I have that is an incandescent bulb a tube light. Now it may be visible to you through air over let us say 10 meters sometimes you know I mean if there is no obstruction may be over a kilometer you can see light. But the intensity it becomes reduced as you go distances because you know that it goes as inverse square. So after some distance the intensity of light essentially becomes so dim that it is of no use at all. Now coming back to whether the light can be used to store information the answer is yes because what I could do I could keep on switching the light let us say I switch it in with certain frequency and decide that this is the way the carry is there. Next question was if air was not good you know physics people have been using the light in the laboratory for transmission and they use prism lenses and things like that. But the problem is their distances over which the light is carried in the lamps is small. So therefore, they use air that is if you are doing a lens experiment in the lab you are still using air. The so the question is that the electrical engineer started asking this question can we use glass you know some or other I can make tubes of glass can I can I transmit it? Now that is where a bigger shock came to our engineers that while physics people were restricted to using you know prism lenses over distances which are the laboratory distances few meters. The what we require is to send signals over kilometers if not thousands of kilometers. Now it turns out that glass is extremely a lossy medium and typical loss in the glass medium is 1000 dB per kilometer. Now you must I remind you that the dB is a logarithmic scale. Generally if 10 dB per kilometer means an attenuation power by a loss of attenuation of power by a factor of 10 in travelling over 1 kilometer. The 20 dB per kilometer means it will lose a factor of 100 because it goes by a logarithmic factor and so you can see the amount of loss that I will have if I am to talk about 1000 dB per kilometer is a hopeless situation. That is even if you could somehow or other technically mold glass into tubes I will not be able to use glass as its loss is fantastic. But while people since people were interested in glass people were doing experimentation and they found something interesting. They found that the loss that occurs in glass is not intrinsic to glass itself. It is primarily due to the impurities which are embedded inside the glass. They would say then how are the physics people doing it? You see the physics people were not interested in long distances. They were only interested in short distances within the laboratory. So therefore if they used ordinary impure glass they will feel no great loss of intensity. But the moment people realized that glass is not intrinsically impure but is impure due to the inclusion of impurities there they immediately got down to doing purification of glass. And what they found is this? Two things. These are engineering problems. Two things happen. One is they could find that the glass could be purified this is primarily chemical engineers and chemists problem to bring down the loss to 20 dB per kilometer which is still pretty bad which is reduction of power by factor of 100 in traveling over a kilometer but I will come back to that question. And then engineers were able to have the glass molded into thin fibers and that is what is known as an optical fiber. So with these two things that is mechanical way of engineers and electrical engineers combined doing the optical fiber and the chemist chemical engineer material scientist physicist all combining to bring down the loss of gas to 20 dB per kilometer that we have now a basic possibility of optical communication. Now what about the loss of 20 dB per kilometer? It is pretty large still but if you look at the loss that one gets in coaxial cables and the your waveguides then you will find this is comparable loss. So in other words I increase I do not get much better quality by this but on the other hand I use my increase my channel capacity substance in it. Now come to whatever the source now as I said that I do not want a steady source I want to be able to modulate that source and how is an electromagnetic wave modulate electromagnetic wave has three characteristics amplitude, frequency and the phase. Now in order to be able to use light as a means of communication of coding information I should be able to switch either the amplitude or the frequency or the phase as a function of time. But problem is the ordinary light like what we have in our homes in kandescent bulb you cannot switch on at very fast rate and the reason is the rate at which you switch it depends upon your spectral width narrower the width the better is the rate of switching higher is the rate of switching. And normally the bulbs that we have at our homes etcetera they are generally wide specter they have wide spectral width. So, they are not good and that is where laser comes into the picture. Now my next few minutes whatever time I have today I will be talking about lasers that what is a laser what is the principle of laser. Remember I am a physicist like most of you are. So, do not ask me the questions which are more proper for electrical engineers or people who fabricate lasers etcetera I am not capable enough doing it. So, let me let me explain what is a laser we will talk to its principle little later a laser in one word of course, you will immediately start saying light amplification by stimulated emission of radiation everybody knows that, but that is not what I am asking. I am asking a question as what is a laser a laser is basically an optical oscillator. Now this is a typical oscillator it has an amplitude it has a time variation and I keep on having a wave which keeps on. Now suppose I had an oscillator which could give me this type of a thing that is a wave whose which extends from minus infinity to plus infinity goes on forever. If you do a Fourier transform of this you find that the corresponding frequency is a sharp single frequency with 0 width. In fact, the Fourier transform of this function is just a delta function in frequency domain. So, if I have a continuous wave extending from minus infinity to plus infinity continuous there in time then I have a delta function that is a frequency at a particular sharp frequency and with no width at all, but that is of course, impractical. I said laser is an optical oscillator I am first let me explain what is an oscillator. So, what we do is this that what is a typical oscillator. So, normally however, if I have an oscillator the oscillator is not this picture that I gave you beautiful picture which extends forever. So, that if I look at it in the frequency domain it is a sharp line. The oscillator I have in practice are those whose amplitudes keep on reducing there is a damping effect and you must have seen this picture of damped harmonic oscillator where the envelope of that function goes at e to the power minus t by tau. Now, if you were to look at it in the frequency domain then you find now it has developed a width. It has developed width because and which depends upon this gamma factor or the factor tau. But before I do tell you what is a laser oscillator let us look at some of the familiar oscillators that I have got. So, firstly this is a normal pendulum and I know that I can set it into motion the time period is given by 2 pi square root of l by g l is the length g is the acceleration due to gravity, but the type of if you leave it like that it does not give you oscillation forever. It in fact gives you damped harmonic oscillation because of the fact that the damping occurs due to many things one is the air second one is the reaction at the hinge and things like that. So, the oscillations usually died down and the this is for example, a 1 1 dimension oscillator mass spring system the time period is 2 pi square root of m by k and same thing if you let it oscillate it will dump and go away. An example of an electrical oscillator is an l c circuit whose time period is 2 pi square root of 1 over l c. Now the question is this that how do I go from here this picture to this picture because this picture has a wide bandwidth and I am interested in a bandwidth like this is possible. The so the way to do it would be that as the oscillations are dying down I must some other supply energy from outside there is no other alternative and for example, I could do that in the first 2 cases by essentially forcing them by mechanical forces from outside that is provide extra mechanical energy. In the in the last case I could simply continue to have oscillations by using an electrical amplifier, but this is this is roughly what one has to do that continue to supply energy from outside. So, we need to continuously supply energy. So, my problem as it seems to a physicist has now boiled down to 2 things I need an oscillator which I have called as a resonator. I need a method of overcoming the losses and that is I need to have forced oscillation maintained. Now let us look at what is what happens in case of a laser the reason I talked about this principle here just now is that the principle of laser is nothing different laser is an oscillator accepting that it is a an optical oscillator and let us see what happens. Now what is the basic idea of laser is this that I have a 2 you know this picture is very similar to what you get for a stretch string that is also an oscillation which I will come back to. So, I have a highly polished mirror at one end I have a partly reflecting mirror at one the other end. So, that normally what will happen is that forget about this red pink thing at the moment that supposing I had a ray of light going there it would be reflected coming back go back reflected coming back and part of it because it is partly reflecting will also be transmitted that will be my output beam, but as I have seen that I need to continuously supply energy and that is where I do what is called pumping I supply energy by and this energy is supplied to a medium which goes by the name of amplifying medium or a gain medium. So, this basically this along with this pumping process is my method of providing energy when the energy is getting reduced here. So, let us look at you see this picture is the same picture that you must have seen in transverse oscillation of a string, but it is no different the thing is this that I am even in this case I am doing the same thing I have a transverse electromagnetic wave which is going from 1.20 if you look at that the way the wave lengths we know that these forms standing wave pattern. So, the smallest wavelength that is possible is when one node is here another node is there and if the length is l. So, basically in the wavelength is lambda 1 then the wavelength lambda 1 is 2 times the length l because this is just half wavelength and the corresponding frequency is C by 2 l. Now, come to this situation next mode the this mode this is called fundamental the first overtone the first overtone is when the node is in the middle there is a node here. So, that the length l and the wavelength in this case I have called it lambda 2 they are equal the correspondingly the frequency has doubled this is this is this is your length l and the your essentially l is the total l is lambda in this case again I have this one the correspondingly the frequency is this. So, what we are trying to say is this that supposing I have fixed a length l now I now look at what are the various modes that are being generated look at this this is that this is that this is 3 by 2 l is l is equal to 3 by 2 lambda and things like that. So, suppose I have this type of a situation where I am looking at q th mode when I say mode I mean this is your first mode second mode third mode etcetera. So, I am looking at the q th mode so that my lambda is 2 l by q now suppose I take a 1 meter distance between the mirrors and I have a wavelength of 0.5 micrometer then the corresponding value for q that I want you convert that into frequency it turns out to be 6 into 10 to the power 14 hertz and I have come to the optical regime. So, what we have now decided is that the laser basically requires to have one highly polished mirror one partly polished mirror and I need to continuously supply it energy. So, that it you know the oscillations are maintained and if I can excite a very large number of modes then I can go to a frequency which is in the optical regime. Now this problem again is due to Einstein in 1917 he used the concept of he discussed this is normally known as the semi classical theory of a radiation by Einstein the way it works is the following. So, look at what he is talking about. So, I am given you here the typical atomic levels I have a energy E 1 I have energy E 2 I have an energy E 3. Now if I have these 3 levels then I know that if I have a population of electrons for example, in this level then I can come from here to there I can come from third level to the first level I can go from third level to the second level every time emitting energy and of course, the reverse process would be true if I am to absorb energy. Now what Einstein did is to look at what are the possible processes that can happen if I am looking at a equilibrium situation. Now in that context the a factor that is there is known as the Boltzmann factor now which tells you that supposing I have two levels of energy E 1 and E 2 and I am at a temperature T the equilibrium situation says that the number of electrons that can be in the lower level will be much larger than the number of electrons which are in the higher level and the ratio is given by the E to the power minus the E 2 minus E 1 by kT. So, in other words the probability of occupation of a level is given by the Boltzmann factor which is exponential of minus the energy of the level divided by the Boltzmann factor times the temperature T this is known as the Boltzmann factor. So, this is basic statistics that we are doing. So, N 2 by N 1 is given by this now because of this Boltzmann factor if I have a at any given temperature the lower energy states are more occupied than the upper energy state. In fact if you do some calculation you will realize that you see the reason is very simple that our kT temperature is very small normal temperature for example, if you take room temperature the in terms of corresponding energy the room temperature is just about 1 over 40th of an electron volt that is 0.025 me electron volts and if that this separation between the two levels is let us suggest even one electron volts then it means that in order to add thermal equilibrium one of the levels will be 10 to the power 40 times more occupied than the other one. So, that is the primary problem e to the power 40 actually it does not matter really. Now let us look at what is happening now firstly let us look at what Einstein call as the spontaneous emission probability. Now spontaneous emission happens if there are electrons in the higher level because they do not like to stay in the higher level they will always like to come down because lower the energy better is the problem you know I mean the chances of the electrons being there is better. So, Einstein defined a coefficient which he called as A 2 1 it is called Einstein's coefficient. So, the number of spontaneous decays per second is simply proportional to the number of particles in the higher level nothing else it depends on. So, I write down the number of spontaneous decays per second is N 2 into A 2 1. So, that was spontaneous emission probability now there is an induced transition which is possible that is the transition that takes place in the presence of a radiation. Now in that case I will still have a transition from 2 to 1 and that transition probably coefficient is called B 2 1 the Einstein's B coefficient, but simultaneously because the a radiation is present the levels can the electrons can actually absorb energy from the levels and also go up. So, there is an upward mobility probability of 1 from 1 to 2 which is B 1 2 and the transition probability from 2 to 1 is B 2 1 there is no great detailed physics being done here, but this number this is the probability this number now depends upon not only the number of particles which are in these levels, but also on the energy density which is there. So, which I have written as U nu. So, there is a difference between the spontaneous and the induced the spontaneous one simply depends upon the number of particles in the higher level whereas, the induced one they also depend upon the radiation density that is present there. So, what happens is this that if you have an equilibrium then the number of upward transition must be equal to the number of downward transition. So, this is from 1 to 2 you remember I wrote down and this is from 2 to 1 2 to 1 had 2 contribution a spontaneous one and an induced one. So, if these 2 are to be the same and it is the Boltzmann factor which governs them you just do that and you find out that the you know equate these 2 and you can get it that I must have a relationship of this type, but you must have also done a Planck's radiation formula. Now, in order that this thing that I have obtained is consistent with the Planck's radiation formula what is required is that B 1 2 equal to B 2 1 A 2 1 by B 2 1 is equal to this this factor comes from Planck's radiation formula and. So, therefore, when the atoms are in thermal equilibrium the ratio of stimulated emission rate to spontaneous rate B is this. So, this is just your factor by which they take this. So, if you are looking at ordinary optical source in visible region for example, the spontaneous emission always dominates and the spontaneous emission is totally random the source is not coherent. Now, in a laser what happens is that the we we build up a certain preferred mode I will just come back to that question in a moment because I do not want to do any more mathematics here ok. And it builds up to a larger values such that the induced transitions become dominant and this thing because you have induced by it by from outside the corresponding emitted radiation happens to be highly coherent. And the spectral radiance irradiance is much greater than that of ordinary light and that is my laser. Let us look at what is it that I I need in order to have a laser. So, notice one thing firstly the first basic point is Boltzmann distribution. We said that the lower the energy higher higher is the probability of occupation and this curve this curve has is an exponential curve this is the Boltzmann distribution. Now, in order to achieve induced emission or stimulated emission what I require is there must be more number of particles in the higher energy level than in the lower energy level. This thing is known as population inversion population inversion because normally if you just let a system gravitate towards spontaneity what it means is there will always be more particles in the lower level than in the upper level and whenever there are particles promoted to the upper level they continuously try to conduct by emission. Now, the question then is this that how do I ensure that the number of particles in the higher level is lower than the number of particle in the lower level obviously it is normally not possible. I want the rate of downward transition to exceed than the rate of upward transition. So, the first principle of again medium or amplification is that the light beam will increase in intensity that is it will get amplified as it passes through the medium that is because this stimulated emission exceeds the loss due to absorption and as we said that this induced emission happens to be in the same direction as the primary beam and as a definite phase. So, this is my situation that my laser basically is I have a collection of atoms in this gain medium molecules and ions in whatever phase and I have an output and then E I out is multiplied by a gain factor. I will conclude this with telling you what is the basic principle. Let me look at first least let us suppose I have a situation where I have just you see atoms have various energy levels and what makes some of the atoms more suitable to become laser material is there where are their various energy levels located. Suppose when I just had a two level system forget about this picture this side of the picture. Now, then I have N 1 and N 2 that a normal temperature the population in the level N 1 is very high and N 2 is low. Now, what you could do is to say all right I want to achieve population inversion. I will pump by supplying energy electrons from here to there it goes there, but you see the problem is it will very soon drop down the moment you stop the pumping it drops down because the life time in the excited state is much lower than the lifetime the ground state it atoms likely to be in the ground state. However, if I had a third level here then what I could do is if somehow I could put electrons in this level. Now, there would be electrons coming back to that level also, but there will be some electrons which are going here also. Now, if it turns out that the life time here is higher than that in there which is very common then there will be build up of electrons in this level. So, this comes here stays there for some time before decay. Now, this actually requires a fairly high pumping level pumping this is called pumping pumping power and that is why a real laser system use as a 4 level system. So, you have N 1, you have an N 2, you have an N 3, you have an N 4. Now, these are various lifetimes you have a T 4 3 is the lifetime of those of transition from here to there T 3 2 from here to here T 2 1 from here to there. Now, suppose this lifetime is large compared to this lifetime then a population inversion will occur between 3 and 2 because this is shorter and this will these 2 levels will providing the laser and lasing levels. There are many methods of doing excitation the electron excitation in elastic atom collision and things like that, but this is the basic principle of laser that is that 2 or 3 things 1 is you have an optical cavity. What was the role of the optical cavity? Optical cavities role was to provide a method of selecting modes. Now, see remember I told you that depending upon what is L, I can have various modes generated. So, let us suppose these are the various modes generated I have already told you that they are roughly equally spaced. What happens is that supposing this represents my loss line. Now, I want I will have lasing provided that is I will have the cavity resonant provided I can overcome this losses and in order to do that I need a gain medium. Now, it turns out that the gain medium has a frequency response. So, if for instance my gain spectrum was like this this is the frequency then the amount of power that any mode is picking up is still lower than the loss line and therefore, no lasing will occur. On the other hand suppose I did this that supposing I had gain medium which is like that then you notice that I have picked up 2 modes and then for these modes there would be laser action. So, this is the basic principle of laser and as I said the there are 2 basic ideas provided by physics into this the principle of critical angle total internal reflection because of which the optical fiber works and the second one is the optical oscillators which is why the source works. Detector is also has some physics in it, but both both because of loss of time and less amount of interest in it for physicists I will not be discussing the your source detector at all. So, since this is the last hour I will be not in a position to directly answer many questions, but whatever people are asking please send the questions on the Moodle as you have been doing and this time I will not be able to spend some time in the morning to reply to those, but I will be replying them on Moodle itself, but let us see a few questions techno India Salt Lake Kolkata yes. Good evening sir. Good evening please go ahead. Actually in your slide 28 number slide. Yeah I know there is a frequency yeah yeah there is there is an I pointed out there is an arithmetic error in that. Longitudinal modes. There is an these are all longitudinal mode I have I know, but I have I have illustrated it for stretch string where the vibration is transverse you know. So, this is not a longitudinal mode really. See if you take a stretch string your direction you see that is not a. Actually for a stretch string. Yeah. I want to know that for longitudinal mode the mode actually depends on the on the length of the cavity, but for transverse mode what is what is the factors. See I was illustrating it with a stretch string a a we we can see I was I was giving an illustration using a stretch string in which I can excite transverse modes because I can pull the string down and the particles motion is in the transverse direction. And so that is how I showed done those. So, I have a couple of questions regarding coherence. Yes. So, can you please go to slide number 2. So, in the second point here you have told that they are incoherent if there is a random or changing phase relationship. Yes. My question here is this changing phase relationship if it is well defined. Yeah. Do we consider them to be coherent or does the phase always have to be constant. Okay. Make the second question I will answer it together. So, the second question is related to slide number 8. So, over here if you do not mind can you please elaborate upon the condition for them not being considered coherent. That is you have said del phi is almost equal to 1. So, how does this become the condition for cannot be considered as coherent. Can you please elaborate on that. Yes. So, coming back to the see the point is that when I said that there is a phase you see we very frequently say phase should be phase difference should be constant. But however supposing the phase difference is well defined function of time I will still be able to control that. You see basically what we are trying to say is this that we if we have two random phases interfering then when I take a time average remember my interference term had that e 1 dot e 2 and I said about because of the fact that my detectors have a response I need time average. Now if I have a random functions when I multiply them their time average is obviously equal to 0. Now if the phase reference is changing but it is changing in a predictable way then I will be able to find out what is the contribution to a to the of the interference term. So, interference will not be stable but on the other hand there will be interference which might be pattern may be changing with time but I will still be able to see an interference. So, I 1 I will not be equal to I 1 plus I 2 as long as that the time difference that the interference that I have got its time average can be calculated and it can be shown to be constant over time. So, that is question number 1 the second one I told you is an empirical law which the optics people have said that what should be the phase difference you know we said that the phase they should be in phase. Now if I have different wavelengths then I have their wave vectors which is given by 2 pi by lambda 1 2 pi by lambda 2. Now the people you know we very often need to say that if the phase difference is so much then I say that the phase coherence is not maintained I mean and that is an empirical relationship. So therefore and that is where they said supposing the phi between 2 sources the phase between 2 sources differ by you know I mean as much as 1 radian then you say the phase coherence has been lost. And in fact that is a decent way to look at it I am not saying whether it is 1 or 5 or 10, but what we are trying to say is that as you go away on your screen your difference in the path that increases on the screen you know you go to fourth fringe fifth fringe your path difference is increasing. Now the reason why the interference pattern will vanish after you have seen 3 or 4 changes is primarily because you are probably not seeing the same wave in. So you say their question of pure monochromaticity does not arise, but if there is a width of your spectrum you convert that into the frequency domain in the frequency domain there is a width which is this essentially C by delta nu. Now what we are trying to say is this that since there is no source known which continuously emits the same wave because if you have a mixture of waves then of course you need you have a width and so that statement that I gave on slide number 8 is actually an empirical relationship that if that happens then assume that phase coherence is not there. The radiation coming from the stimulated emission why it should have the same phase relationship with the incident radiation why the two radiation should have the same phase relationship. Yes. So the idea is something like this that the incident radiation is what you are controlling because that is what is pumped into the system from outside. Now that is what is its energy density is responsible for giving you that some things are going up you know in the presence of that. So basically what happens this is what is found the actual reasonings are little more difficult to explain but what I have what was found or let us say Einstein's theory is a semi classical theory. So what he assumed is this that because you are forcing the process. So this radiation that takes place in the presence of a motivating factor depends upon the motivating factor I mean as to why exactly it happens I think there are now theories of laser which try to explain that but I do not know much about it. Thank you let us stop okay we will conclude and go over to the next.