 So, today we go a little further without discussion of how to actually do mode locking, so far we have studied the theory, today we are going to talk about at least one case in which mode locking is done and how it is done. If you read any standard textbook at this stage what people discuss is active mode locking, passive mode locking and acoustic modulators, we will take a rain check on that, we will not do it right now, because our primary goal is to understand how a tie sapphire laser works at this point of time. So, we will skip that for a moment, we will come back and do it in the module after next, after we have shown you tie sapphire laser. But today we want to learn about a very strange way in which mode locking gets done and femtosecond pulses are produced in titanium sapphire laser and the way it is done is called curl length mode locking. And in fact the curl length mode locking happens in a manner in which you do not even perhaps understand that it is happening. So, it is almost like it happens by itself. So, the question is what is this magic, what is the mystery by which this femtosecond pulses that we are looking for so desperately get produced by themselves without us having to do much. And is there anything that we can do to improve the situation that is what we will discuss in this module. In the next module we will go and talk about an actual tie sapphire laser. But before that let us quickly recapitulate what we have discussed in the last couple of modules. We have talked about longitudinal modes in detail. So, now I hope we all understand that a laser can have many modes, many modes mean many frequencies that can be sustained in the cavity and the kind of cavities we have and the kind of color of light that we deal with, we always deal with things like 10 to the power 6th, 10 to the power 6th mode, 10 to the power 6th plus 1th mode and so on and so forth. So, these modes are only slightly different from each other as far as frequency or wavelength is concerned. So, and we have also said that this spectral width of each mode is something that we have also calculated. And given the modal wavelength and given the width of the spectrum we know what the total number of modes is. And then we introduced another term which is very important as far as discussion of laser is concerned not just ultrafast lasers any kind of laser that is quality factor. Quality factor is essentially a weighted ratio of energy gained in the system and energy lost during one cycle. Energy gained in the system means what is the build up how much a population inversion is achieved and that is usually achieved by pumping it with an intense laser and energy lost can be not just by emission of light but also due to things like heat and so on and so forth. And then without derivation we have given you the relationship between the spectral width of each mode and the quality factor very soon we will come to a discussion of what is called Q switching where we can generate pulses not very short pulses but pulses nevertheless by switching the quality within the cavity. And then this is a figure that we have shown maybe 3-4 times already what happens when you have many modes first of all when they have no face correlation second even they do have face correlation. And what we have said is that when the phase relationship is a function of time then you get a random fluctuation and essentially you get a CW output. However when you have a phase relationship which is independent of time then you achieve what is called mode locking and that is when you generate pulses and while talking about pulses we also said that you get the intensity by taking square of the field. So when you combine N number of pulses this is something I do not think I said earlier when we combine N number of pulses essentially the intensity at the peak turns out to be N square multiplied by E0 square where E0 square is the maximum amplitude of field associated with the normal modes. So you see you couple 5 modes let us say E0 is same you couple 5 modes you will get 25 multiplied by E0 square you couple 500 that will be 500 square multiplied by the same E0 square. So that is how peak intensity grows very significantly as you increase the number of modes that are coupled. And we said that this is the shape of the intensity of each pulse if you zoom into it you have one major pulse and you have very short side bands and the widths are very very small here. We also talked about trepidation rate turns out to 12 by C which is essentially round trip time within the cavity of length L. Pulse width however turns out to be something that is dependent not only on the length of cavity but also on the number of modes that are coupled together. And since we know the relationship between number of modes and lambda 0 the modal wavelength and delta lambda we worked out this relationship that pulse duration Tp is not really full width or half maximum pulse duration turns out to be lambda 0 square divided by 2 C delta lambda. And this is something that I have not shown earlier this is so you can see when this was published 1964 in the very beginning of the invention of lasers itself. So this is what you get this is what I was talking about you couple 5 modes these are the kind of pulses you get quite broad and less intense. You couple 50 modes you get sharp and intense pulses right. So this is the essence of mode locking and production of pulses. Let me say that the best case scenario you can get is when you have transform limited pulses. So product of pulse width full width or half maximum and product of full width of half maximum of the spectrum turns out to be a constant which is 0.441 for Gaussian pulses and different numbers for pulses of different shapes. But remember this is the best case scenario you can get just because you have a certain bandwidth need not necessarily mean that you have a pulse width that is as small as you really wanted to be. You really have to have good alignment in order to achieve transform limited pulses. Before we move on it is time to talk formally once about transverse modes. What is a transverse mode? You take a laser beam and put it on a surface may be expand it and put it on a surface. You are going to see a spot the question is what does this spot looks like? If the spot looks like a circle roughly a circle without any dark spot anywhere it is a patch of light circular patch of light then this mode is called tem00 mode. Tem means transverse electromagnetic mode T for transverse T for electromagnetic M for mode tem00. 00 means there is no node in any direction this one is called tem10 mode that means there is one node in a particular direction. This is 11 mode one node along this direction one node along this direction. Now the problem is this here is 10 mode this one also has one node this is called 01 mode which one is 10 which one is 01 there is no good answer to that because directions are all arbitrary right you might say that this surface is XY so this is X axis so if there is a node along this I will call it 10 if there is a node along this direction I will call it 01 but then X and Y they are all relative right it depends on how you define the directions so as EW small says in this chapter of JR Lacovitch's topics in fluorescence spectroscopy this is not the introduction to fluorescence spectroscopy book Lacovitch also has several volumes of topics in fluorescence spectroscopy published long ago there in this chapter 2 EW small says one man's 1 0 mode is another man's 01 mode so it depends you look at different books the crux of the matter is this number 1 or 2 or whatever it is tells you how many nodes there are in a particular direction now which direction comes first which direction comes second is absolutely a matter of convention okay this one is 4 0 or you might want to say 0 4 mode before coming to this one do these modes remind you of something that you have studied in your first year BSE or something yes the reminders of orbitals right the shapes are very much like orbitals in fact when I taught this course actually in a classroom one of the students sent me this pictures of orbitals and said do not they look very similar actually yes they do okay now the last so you can go on and on these modes what about this is there a node here you have what looks like a 0 0 mode but then there is a whole pulse in it this is called in colloquial terms or do not mode 1 0 star now one thing I have not drawn here is you can also have nodes that are radial right so here I have drawn nodes as planes but you can also have a spot then a circle of darkness and then another spot of light on you got 2 of them 3 of them 4 of them so radial nodes are also there again you call them 1 0 2 0 so on and so forth now this do not mode 1 0 star is of particular importance in things like super resolution microscopy so in I do not think we will have scope to discuss super resolution microscopy in this course maybe sometime later so I think we know right what is the smallest spot that I can get by focusing light of wavelength lambda lambda by 2 right why is that so it is called the refraction limited spot so you cannot make it any tighter we can think that we will do tight focusing and the spot will become smaller smaller smaller but then it cannot become 0 will always be a finite size because below that diffraction will set in and brought in the spot a little bit so diffraction limited spot so if I use what kind of I want to do say visible microscopy visible light microscopy 600 nanometer 500 nanometer something like that let us say I am using a 500 nanometer laser to do microscopy so then the best resolution I can get is 250 nanometer right so you cannot do any better by usual microscopic techniques as you know I think 2014 Nobel Prize was for super resolution microscopy who are the people who got it Stefan Hell was one Bedsig was another one and Mohrner was a third Mohrner actually got it not for super resolution microscopy but for single molecule microscopy and spectroscopy Bedsig and Stefan Hell got Nobel Prize for super resolution microscopy super resolution microscopy means doing microscopy using visible light with a resolution that is better than diffraction limit how do you do it there are Stefan Hell method is you take two pulses one with a 1000 mode another one with 10 star mode and you overlay them and then you can play around with this and have destructive interference between the two pulses at the instance when you have destructive interference what happens is this wherever there is light in this 10 star mode in the composite beam in the composite spot that becomes dark so the beam becomes smaller smaller in radius okay so of course when I say radius what do I mean by radius will come to that so 10 star do not think it is just for our purpose when we do ultrafast spectroscopy in bulk mode we do not want 10 star mode we will see why but it is useful in super resolution microscopy but for now let us only focus on this 1000 mode in 1000 mode if I want to plot intensity this is what it looks like so intensity is maximal at the center and it falls off on the two sides so in the best case scenario you have a Gaussian kind of distribution my diagram does not look Gaussian that is due to my inability to draw very nicely but in 1000 mode you do have Gaussian distribution of intensity specially and now when I say Gaussian is it two dimensional or is it three dimensional yeah there are two axis right when you take a spot you have one axis and you have another axis here so it is Gaussian with respect to this Gaussian with respect to this also so start at the center intensity falls off as you go out radially okay so this is your 1000 mode and you will see why we are suddenly discussing these they actually useful in card lens mode logging now let us talk a little bit about car effect car effect says that if you have an intense beam then it can modulate the intensity of the medium on which it is incident and without going into further this is a non-linear optical phenomenon we are going to talk about non-linear optics a little bit later on but today let us just take it axiomatically the refractive index of the medium on which an intense beam falls changes in this way n0 is the refractive index when the an intense wind is not impinging on it when it falls then there is a second term which depends on the intensity of the beam n2 into I now one thing one needs to be careful about is n2 is not what is the unit of refractive index refractive index no unit right it is a ratio but do not think that this n2 just because I have written it n2 do not think it is devoid of unit you see left the equation has to be dimensionally consistent right and here it is not just n2 multiplied by intensity intensity does have a unit so n2 has the dimensions of inverse of intensity okay now see what happens you have this Gaussian beam I hope you all understand contours right the as the contours get smaller that it represents the we are going to the peak higher intensity so as this is the center of the beam as you go out what will happen intensity will fall off in this manner as we have discussed right and this is from a printed book this is a prettier diagram than I could have drawn what will happen to in the refractive index intensity is maximum at the center right and it falls off on the two sides so what will happen is for a sufficiently intense beam the refractive index will also fall off in the same manner refractive index will not be the same across the beam profile in the media are we clear is there a question we are going to have this kind of a Gaussian variation of refractive index in that media this is called Kerl Lensing why is it called Kerl Lensing because think of a lens think of convex lens how does it work for a convex lens the any light impinging on it goes through more of the medium right so more refraction takes place here we do not have convex or concave medium we have let us say a cubical block but we due to intensity of the beam we are bringing in a gradient of refractive index so effectively this block we said no reason to act as a lens is now going to act as a lens are we clear about that in this region refractive index is less in this region refractive index is more so what happens okay let me put it in another way what happens when a parallel when a collinear beam of light goes through here it goes through a region of smaller refractive index or rather smaller change in refractive index from what it actually is so it will more or less go straight as you go inside what will happen it will bend more because refractive index is more so eventually what will happen is this beam that was going straight will get focused are we clear what kind of a beam will get focused and intense beam only this is the tricky important part here if you take torsilite make a collimated beam out of it and make it fall on a medium you will not because it is not intense if you take the output of say helium neon laser put it on a medium you will not see carl ensing because it is not intense you take a continuous wave tie sapphire laser make it incident on this some medium you will not see carl ensing you take the same tie sapphire laser button pulsed mode you are going to see carl ensing because remember what intensity is it is n square multiplied by 0 square right mode the number of modes better it is okay so mode lock laser mode locked light is what is going to produce this carl ensing effect okay so an intense beam will get focused and a weak beam will not get focused because do not forget the beam itself is bringing in carl ensing okay this is the part that we really need to understand it may not be so easy to follow when you hear it for the first time and then it is going to get more interesting as we go further so what we have said is pulsed light gives you intense beam so in a mixture of pulsed light and and continuous wave light what will happen you have a medium through which your pulsed light and mixture of pulsed light and continuous wave light of the same wavelength range is going through the pulsed light will get focused and the CW part will not get focused right so if you show it diagrammatically but before that this is just for the record again since you are not deriving it is not much of a fun but then just for the record the focal length of a carl ensing is given by W0 square where W0 is the beam waste beam waste once again is full width of maximum of a Gaussian beam by 4 N2 I0 multiplied by L where L is the thickness of the medium length of the medium so now with this understanding we suddenly have something called carl ens mode locking what is that about let us see what did we say we said that your induced refractive index depends on the intensity so we are starting with this that we have a mixture of CW and pulsed light okay in a laser somehow so we can think like this you have started a laser and some of the pulses have got modulocked may be 5 pulses and there are other pulses that are not modulocked this modulocked pulsed light is going to be intense so you will get focused okay now think that we have a laser consisting of your ticep fire crystal nothing else titanium sapphire crystal and 2 lenses we will introduce something very shortly that ticep fire crystal itself can act as a car medium is it light is going through it it is a solid crystal so any light that is passing through the ticep fire crystal itself is going to show this carl ensing effect okay so this is what will happen your mixture of CW and in fact it is not a simple mixture of CW and pulse maybe there is a gradation so intense light will get focused like this and CW light which is not so intense will not get focused or will get focused to a laser extent okay so the dotted lines denote the CW light the solid lines denote the intense light is there any reason why the solid lines are inside and the dotted lines are outside do not forget that it is a Gaussian beam so intensity is actually maximum at the center okay so it is a sort of a synergistic effect so you are going to have this kind of a car medium your pulse light will get focused CW will not get focused now if you introduce an aperture here what will happen see this one near M2 here these black lines denote the section of an aperture pinhole the CW light in the fringe will get blocked and pulse light more towards the center will pass through okay so combination of car medium two mirrors and a pinhole can give you a laser whose output is pulsed any question so what I am saying is this you understood that this titanium sapphire laser titanium sapphire crystal that itself can act as a car medium right you are pumping it and then you have this light passing through it the stimulated emission to start with let us say some modes are blocked somehow that will happen right it is there is always a possibility of happening so those that mode locked beam of light is going to get focused and unmode locked CW light will not get focused now if you introduce an aperture here then the CW light which is on the outer side that is going to get cut and the intense mode locked light which is at the center will not get cut so it will go through so only the pulsed operation will be sustained CW operation will not be sustained okay now let me say something more is that aperture even required to understand to start with it is good to have the aperture there suppose I do not have an aperture will this thing still happen so you have curved mirrors right so something that is focused and something that is not you make the cavity in such a way that you have curved mirrors which support this kind of focusing here so anything that goes straight will not come back here so it will automatically go out of the cavity remember the terminal mirrors are never plain mirrors there are always curved mirrors so in fact that aperture is not even required you can get pulsed operation without the aperture also okay this is called curl lens mode locking now curl lensing gives you a pulsed operation that is great but it always makes things a little it also makes things a little difficult for you because it introduces what is called sharp what is the meaning of sharp before that let me say that these are nicely animated slides that you are going to see I did not make them these are made by Dr. Johor Alam Mondal of BRC many many years ago she was kind enough to give me the slides so I use them so the thing is this in curl lensing you have this pulsed slide incident on it I hope it is not very difficult to recognize this as a pulse I am drawing the electric field not the density then it induces curl lensing okay so effectively even though it is not even though it is just a slab as far as the light is concerned the slab behaves like a convex mirror like this okay what you see here is sort of a heat map for your refractive index becomes a convex mirror and then focusing takes place all that is good but the problem is what is the meaning of drawing a pulse like this it is a mixture of many modes is not it and more modes you have the better it is so you have different frequencies and you have a lens what is a common problem that is encountered when you have polychromatic light being focused by lens in secondary level optics different colors there is a hint the problem yes what different the speed is different exactly so what happens and what is that called it is called chromatic aberration and you corrected by using a chromatic doublets right so that sort of negates chromatic aberration so here since there are so many different wavelengths involved do not you think chromatic aberration will happen yeah so and as a result of chromatic aberration what you will get is you will get a chart pulse some color will go ahead some color will go behind and that will cause a broadening of pulse so before you can actually start making a laser one needs to correct for this chromatic aberration ok once you correct for charting how we will do that in the next module