 Now we will begin our discussion on longitudinal modes but before we do so we have a question what is the question? So question is how to find out peak power? To find out peak power it is important to realize that what we see here it is the power or energy per pulse it is basically area under the curve. So from there suppose the pulse is triangular it is easier to understand triangles that is why I am saying triangular we will come to Gaussian suppose it is a triangular pulse then your area is half into base into altitude. So from there if you know that we had taken 200 femtosecond to be the width that is the base you can find out what altitude is and here the point to remember is that the base is really very small. So half into base into altitude base is small so altitude will automatically become very large. And now for a more realistic situation where we have a Gaussian kind of distribution there what we will have is we are not going to have half into base into altitude but even Gaussian has a line shape right it is I0 intensity at the maximum position multiplied by a Gaussian distribution e to the power minus x square by 2 or something like that. So from there we will still be able to find out I0 that is how you find peak power and for small pulses and same kind of energy peak power is really, really high. Damage threshold is usually peak power not average power because damage is a result of one to one interaction of light and molecule right. So sometimes what happens is that you know what kind of laser you are using so in that case your peak power is proportional to the average power and proportionality constant is known the system is known then average power may be mentioned. But what you really have to worry about is if your short pulses peak power is the killer if you are worried about damage of your system sometimes damage is good because you might want to do lithography or you might want to do laser cutting or something like that in a favorite example of mine nothing to do with spectroscopy to be honest I do not exactly remember the year most likely sometime in 2003 or 2004 or something like that there was this interesting paper in which somebody had manufactured a bull bull the animal that is rampant in our campus bull a he cow a bull. Now of course being in IIT Bombay one might wonder what is the need of manufacturing bull there God has manufactured plenty and they have provided plenty of problem for us all the time but this bull was special this bull was a nano bull so it would not provide too much of a trouble to you and the way they worked they just wanted to demonstrate there is no reason for making a bull otherwise you cannot even see it unless you put it under a microscope but the way this bull was prepared was that a pulse laser was focused on to this monomer solution and the idea is that the light that you put in it it is going to cause polymerization so wherever the focal point is that is where polymer will be formed and then you move the stage carefully and the polymer bull is formed it is nice very very good nice good looking bull you can read that paper but then the point that I am trying to make here is that it is focused right and you are talking about damage threshold here this is not damage this is the threshold that is required for the reaction to happen and then if you want a very good resolution so here the trick that was played was 800 nanometer light was focused but actually 400 nanometer light was required for polymerization to take place so polymerization took place as a result of 2 photon excitation now 2 photon cross section is actually square of intensity right so it is narrower than the actual pulse that is why better space resolution was obtained by focusing light not of 400 nanometer but of 800 nanometer I do not perhaps we digress a little bit to answer that question but intrinsically they are related something that can destroy can also build if it is used properly right now let us come back to a point of discussion we want to talk about longitudinal modes now you might think what is going on here what is where did this come from we are talking about lasers we talk about pulse lasers we are saying that you packed many photons in all this discussion what is the meaning of longitudinal mode and what is the relevance I cannot say what the relevance is right now you have to wait maybe 15 minutes or 20 minutes to arrive there okay so let us first understand what these modes are as we mentioned in the previous module there is something called transverse mode means specially what does the laser beam look like if you take the laser beam laser spot on a piece of paper do you get a circle do you get a dumbbell do you get something that looks like a d orbital and there depending on the number of nodes along x and y direction you call them time 0 0 time 0 1 time 1 1 so on and so forth modes but right now well they are important we will come to them later right now we want to talk about longitudinal modes longitudinal means modes along the direction of propagation of laser light okay. So now it is important to understand that for a given cavity not all wavelengths are going to be sustained because in order to get a beam where light is going back and forth you must have constructive interference right standing wave has to be generated and condition of standing wave is that the cavity length L this one must contain an integral number of half wavelengths L equal to n lambda by 2 this is very well known I think all of us understand that okay so the point we are trying to make is that you make a laser L is defined you decide already that a particular set of wavelengths can be sustained okay in principle this number of wavelengths is infinite because n can go from what is the smallest value of n cannot be 0 1 n can go from 1 to infinity but then also you have some active medium right which has a spectral bandwidth it is not going to give you 1 to infinity okay. So how many modes are there in the bandwidth that we are using that is something that we will learn first why we are learning all this we will see by the end of this module hopefully. So before going further I want you to calculate this n and the reason why I want you to do this is it is very easy to know thing that okay L equal to n lambda by 2 take an isofi laser since I have used 900 nanometer here I will take that number 900 nanometer is a wavelength that is sustained in isofi laser perhaps that is n equal to 1 so n equal to 2 for n equal to 2 what will be lambda what will lambda be it will be half of that n equal to 3 it will become one third and so on and so forth but that is actually not the case we have as we will see so I we want to dispel that notion that is why we have to once again get a sense of numbers in this business so my question is for lambda equal to 900 nanometer and L equal to 3 meter this is a rough number that I am using here what is the value of n can you calculate yes 6.6 into 10 to the power 6 I have written 6.67 here is 6.66666 6.7 10 to the power 6 that is the first thing to understand we are not dealing with n equal to 1 we are not dealing with n equal to 2 we are not dealing with n equal to 3 we are dealing with n equal to a very very very large number okay if we are dealing with n equal to 1 2 3 we actually have single mode lasers because n equal to 1 and n equal to 2 would differ largely in the wavelength isn't it it will be very easy to cut out one and use another one use an optic optical coupler use use an output coupler that is a dichroic mirror everything other than n equal to 1 or n equal to 2 n equal to 3 would be cut out here however it is important to understand first of all the n value that we use is actually very large okay so if n is equal to this 6.7 into 10 to the power 6 what is the value of n plus 1 7.7 into 10 to the power 6 6.7 into 10 to the power 7 you do not even want to answer that question because it is best to say 6.7 10 to the power 6 in bracket plus 1 otherwise you actually have to write down all those zeros 6 7 0 0 0 0 and then finally 1 0 becomes 1 okay point we are trying to make here is n and n plus 1 okay delta n by n is a very small number delta n is of course 1 when I am talking about n and n plus 1 but delta n by n is a very small number okay how does that matter this is how that matters n into lambda n equal to 2l n plus 1 into lambda n plus 1 is also equal to 2l okay and what we just said is that this n and n plus 1 are practically the same if n is equal to 1 then n plus 1 would be twice n but n equal to 6.7 into 10 to the power 6 that plus 1 is practically equal to 6.7 into 10 to the power 6 if your salary is 1 crore rupees you do not really mind whether 10 rupees are added or subtracted right something like that. So, now what will be delta lambda let us do a rough calculation first lambda into n sorry lambda n minus lambda n plus 1 what will it be it will be a very small number right it is going to be basically 2l divided by n which one will be longer lambda n plus 1 is higher energy right no yes so the difference is going to be not much that is what we understand. So, and that tells us why it is so difficult to make multi mode lasers. So, you have maybe 900 nanometer and 900 and 900.5 nanometer right these are two successive modes how will you separate them ok that is why multi mode lasers usually lasers we have are not multi mode lasers and multi mode lasers are available but they are costly for our purpose this is a blessing in disguise because if you want to do ultrafast spectroscopy you do not want a multi mode laser as we will see ok for now it is important to understand that the difference between the wavelengths of successive modes is really very small right that is best understood if we take this problem of separation in colors of longitudinal modes let me call them colors in terms of not lambda but nu because energy is proportional to lambda right energy is proportional to nu reciprocal with lambda. So, nobody works with delta lambda generally delta nu is better can you tell me what is delta nu when delta nu is nu of n plus 1th mode minus nu of nth mode can you work it out delta lambda equal to nu n plus 1 minus nu n what will it be you have to convert this expression of lambda to nu of course lambda nu equal to c so lambda is c by nu very small is already established I am not asking for that I am asking for an expression I have two answers one is c by 2l another one is more complicated what is the right answer c by 2l right c by 2l delta nu which is the difference in frequencies of two successive modes it turns out to be c by 2l very interesting why is it interesting it is interesting because it is a constant you are working with some laser right l is defined and c is defined anyway unless you change the medium or something so c by 2l is a constant that is the first amusing information that comes out okay delta nu is constant and the only thing that determines what delta how big or small delta nu will be is l right. So, if you want delta nu to be double the value that it is it right now what do you do to the cavity do you increase do you decrease yeah do you make it double do you make it half I want delta nu to become double yeah then l has to become half and I want delta nu to become half that means I want the I want more energetically closer energetically more closer modes then do I need a shorter cavity length or do I need a larger cavity length I need a longer cavity length this is an important issue and this is an important issue for both depending on your requirement do you want more modes do you want less modes as a first control as you see we want more modes if you want an ultrafast pulse so for that you have to increase the cavity length okay where was the first femtosecond pulse produced in India Bangalore is an answer Bangalore no in this case no I gave you a hint I am asking the question now of course it is not our lab it was produced in IIT Bombay it was produced in department of physics professor B. P. Singh's lab he had a CW Ticephile laser so if you go to his lab even now it is there you can see it he made a very strange arrangement he opened up the laser and then he added a vertical inward rod on which he mounted optics that is how he increased the cavity length so you can actually buy Ticephile lasers that will never give you pulse always CW and you find that they are very small our Ticephile laser is first of all big secondly it has folded cavity we are going to open it up and we are going to demonstrate that is because you need a certain cavity length in order to get pulse and then if you want to be a straight cavities simple two mirror cavity then you need a huge space your table is going to be from that to here so it is better to fold it okay by folding is done by using mirrors okay. So C by 12 is a constant for a constant cavity and if you want more modes more modes means smaller separation between modes you want the cavity to be bigger that is the first point what is the second point C by 12 that is a very special number what is C by 12 okay I will make it easier what is 12 by C 12 by C is a round trip time right 12 by C is a round trip time you start you start at this point let us say hit this mirror go back hit this mirror come back that is called a round trip so the total distance travelled by the photon is 2L right and it is a photon so it travels at the speed of light so 12 by C is the time for a photon to do a round trip okay not you or me to do a round trip photon. So C by 12 essentially is a reciprocal of round trip time okay incidental but interesting important. Now I have written in terms of delta nu nu is something that I am not very comfortable with I understand wavelength and wave number better if I want to write delta nu bar what will it be yeah and this is easy question delta nu bar that should not take so much time yeah divide by C what is it 1 by 2L you are hesitant to give the answer because it is so simple but sometimes you get simple answers to difficult questions what is the problem with that so delta nu bar is 1 by 2L okay the only thing that matters is the cavity length now let us once again do a quick bit of math we are working with 900 nanometer let us stick to 900 nanometer can you tell me what is nu bar for 900 nanometer what is the wave number corresponding to 900 nanometer 1 by 900 into 10 to the power 9 10 to the power minus 9 of course okay if I want to do mental arithmetic it is easier to make it 1000 right so 1000 is 10 to the power 3 10 to the power 3 into 10 to the power minus 9 is 10 to the power minus 6 what was that is that right yes reciprocal of that is 10 to the power 6 so that is your nu bar for some nth one nth longitudinal mode what is delta nu bar so nu bar what did we get 10 to the power 6 right 10 to the power 6 what is the unit meter inverse right see once again I was going wrong by a factor of 10 to the power 2 this is my problem so 10 to the power 6 per meter meter inverse and what is a typical cavity length meter 1 meter 2 meter 3 meter does not matter okay we had taken 3 meter earlier is it so let us say 3 meter so what is delta nu bar 1 by 6 per meter is that right if l equal to 3 1 by 12 is 10 to 1 by 12 is 1 by 6 what is 1 by 6 so now you see this is what we are trying to say nu bar for that 900 nanometer light is 10 to the power 6 per meter and nu bar for the next line is 10 to the power 6 plus or if you take on the other side minus 1 by 12 very close okay successive modes are actually very close in energy because we are dealing with high n values we are not dealing with n equal to 1 2 3 are you okay clear and we go ahead alright now we want to know how many longitudinal modes are there why are we doing all this because we are going to see eventually that we want to take these modes and bunch them together to get the pulses okay so you want to know before going there how many longitudinal modes are there to do that let us have all this is from that book whose author's name I cannot pronounce I had shown the book in the during the first module I hope introduction to laser spectroscopy Halina and I cannot pronounce the second name alright here let us consider this this is the laser cavity we have discussed many times here let us say this is the spontaneous emission band intensity versus lambda okay let us say spectral maximum is at lambda 0 and let us say delta lambda is half with that half maximum see what will happen is stimulated emission band is always narrower than spontaneous emission band okay energy density matters is not it energy density of this light is going to be very small energy density of this is maximum so this is sort of like the story of the bull that we are discussing little earlier nano bull okay there also there was sharpening in space because you required a 2 photon cross section here we get a spectral sharpening because of variation of energy density okay so let us say of course this may not be correct for all systems let us consider that this is where onset of stimulated emission takes place okay half with that half maximum of the spontaneous emission band defines the half width of spectral base of this stimulated emission band okay it is not necessary that it is half it can be one third it can be one fourth something but for now let us work with half let us say that this spectral width is delta lambda so what is the base of the spectrum 2 into delta lambda right so once again we are back to the question that we started with but in a different domain that was in time domain this is in wavelength domain okay so now see n multiplied by lambda n equal to 12 fine n plus 1 multiplied by lambda n plus 1 equal to 12 that we know let us say your n max is the mode number so you are giving roll numbers to modes mode number 10 mode number 11 mode number 12 is just that we know it is not 10 11 12 it is 10 to the power 6 plus minus 1 2 3 4 okay so let us say n max is the mode number in this position where will n max be here or here it will be on the lower wavelength side right lower wavelength is higher energy let us say n mean is the mode number for this end of the spectrum okay so we can rewrite these equations then n max multiplied by lambda 0 minus delta lambda should be equal to 12 n max is the mode number here what is lambda lambda is this lambda 0 minus delta lambda so n max multiplied by lambda 0 minus delta lambda in bracket is equal to 12 similarly n mean multiplied by lambda 0 plus delta lambda in bracket is equal to 12 once again okay have you understood what is the total number of modes n max minus n mean well you might want to do n max minus n mean minus 1 okay but then n mean minus 1 n mean are not very different from each other n max minus n mean what is that n max will be equal to 12 divided by lambda 0 minus delta lambda n mean will be 12 divided by lambda 0 plus delta lambda so basically you are subtracting one from the other you will get a denominator of lambda 0 square minus delta lambda square is that right and then you will have lambda 0 plus delta in the numerator you will have lambda 0 plus delta lambda minus lambda 0 minus minus delta lambda so in the numerator lambda 0 will cancel you will be left with yes is this what you have got yeah and what is the denominator lambda 0 square minus delta lambda square but then what we have established so far is that delta lambda is really a small value so you might as well neglect it square for a small value square will be even smaller so what will the denominator be lambda 0 square right so this is my answer 4 L delta lambda by lambda 0 square is this an absolute relation or does it change from case to case see this relationship is obtained with certain considerations first consideration is the onset takes place right at lambda 0 minus delta lambda and lambda 0 plus delta lambda where delta lambda is half with that half maximum that may or may not be corrected can vary it depends on the shape of the spectrum and so on and so forth roughly you get something like this you will definitely get L delta lambda in the numerator you will definitely get lambda 0 square in the denominator that 4 instead of 4 it can be 4.12 or 3.89 it can vary a little bit depending on what kind of spectrum you have and what kind of system you have but roughly this constant multiplied by L delta lambda by lambda 0 square will hold okay now we are in a position to look at this and think what n is going to be will actually work out given some typical values but if L increases n is going to increase okay that may or may not have occurred to us if we did not see it it would not have occurred to me for sure if delta lambda increases and will actually increase I would not have guessed it to be honest and lambda 0 square in the numerator denominator that also would cause n to increase okay if lambda 0 decreases n will increase okay. Now let us work it out for our tie sapphire laser maximum is at 800 nanometer right so lambda 0 is 800 nanometer L okay what is the value of L let us take 4 meter will it help it will help to some extent yes so let us say L equal to 4 meter what is a good value of delta lambda where does lasing start for tie sapphire laser say 700 nanometer let us say 100 on each side okay 200 no 100 right so let us say delta lambda is 100 nanometer can we work out what the number of modes is under that spectrum 4 multiplied by L is 4 meter multiplied by delta lambda is 100 into 10 to the power minus 9 meter divided by divided by is too much what is this divided by what is the denominator 800 nanometer square. So 4 into 4 is good that we will get we will take care of multiplied by 10 to the power 2 square is 10 to the power 4 multiplied by nanometer square right so 10 to the power minus 18 will be there so this 4 is all take care of each other you are left with 10 to the power minus 7 in the numerator and in the denominator it is 10 to the power minus 14 is that right what is the number that is coming how did you get 25 4 L delta lambda by lambda 0 square delta lambda is 100 nanometer so yeah what is your answer because do we get that answer how much 2.5 into 10 to the power 6 large number right so we can expect that for something like titanium sapphire the emission is very high the stimulated emission is very highly multimodal 10 to the power 6 modes are there now let us see what happens in 2 cases one in which these modes are have no phase relationship one in which they have some phase relationship this is what happens here to keep things simple we are not plotting intensity we are plotting the time evolution of electric field so I am showing a large number of waves here okay waves whose frequencies are different from each other just a little so this is a free running laser in which these different modes have no relationship let us say and here this is called a mode lock laser and we will see why it is called a mode lock laser where delta phi equal to constant which means some phase relationship is maintained the easier thing to work out here is at some points let us say at this point phase difference is 0 now what happens you see this kind of a oscillation here and here you see that at this point everything is in phase okay so if this is in phase all the electric fields will add up you will get constructive interference the moment you move x axis is time remember the moment you move out in any direction the waves start getting out of step this is something that we might have studied when we studied time domain spectroscopy right so they start getting out of phase until they have complete destructive interference and then after some time they will start getting back in phase so you are going to get something like these are called interferograms okay so you have packets of energy at regular intervals the interval is shown here we are going to derive this next day but just see what the interval is 3L by C minus L by C what is that 2L by C same 2L by C that we encountered earlier and this is telling us a story what is the story the story is like this suppose you have a shutter in the cavity okay you open it for a very short duration and close it again what will happen in this short duration is that some of these some waves will go through now let us say you somehow manage to ensure that at the time of going through all these waves are in phase okay they come back when they come back to the shutter again you open it what you do then you sustain this mode locked operation rather than free running operation okay so basically bundling together waves that are in phase at that point of time of crossing the shutter and then let them evolve by themselves so that they will go to 0 very quickly and then come back that is how you prepare pulses okay this is only an introduction in the next couple of modules we are going to do the math and then we are going to say how it is actually experimentally done okay that will be our discussion of mode locking if it is not ultrafast then it is perhaps easier to do if you want nanosecond pulses it is done by something called Q switching if it is femtosecond pulses then things become difficult and fortunately as we will see nature provides us a way out by which femtosecond pulses are produced by themselves you only have to put up the system a little bit and the system mode locks itself okay that is what we want to do after this we will we may or may not discuss how to mode lock immediately right away after this but rather after we have developed the concept of how to prepare how to make pulses we want to talk about how to amplify them and then we come to the concept of chopped pulse amplification then we take it from there