 This is where we had reached in the previous module. Intensity of the output beam for a cavity dumped synchronously pumped mode locked laser. I have used several qualifiers, adjectives, well right now even if you do not understand synchronously pumped or if you forget it is okay, but you cannot forget that not only is this cavity dumped, even before cavity dumping it is mode locked. I have said it 3-4 times already, but it is important as you will see all right. So this is the expression for your intensity okay, intensity of the output. So what you see is that this output is modulated by a frequency that is double the frequency of the sound wave used is that right, is that right? The phi's are just phase difference phases, but frequency is given by 2 omega. So it is modulated at 2 omega, 2 capital omega sorry. Now the question is what does this modulation mean? When is this going to be maximum, when is this going to be minimum? Cos 2x, what is the maximum value of cos function cos theta plus 1, what is the minimum value minus 1 or 0, minus 1, minus 1 is less than 0, cos function is not cos square right, it is cos. So your square is on the other terms is 0 square and all, but then it is a cos function. So when will it be maximum, when cos of 2 capital omega t plus phi s plus phi will be equal to 1 and when is that cos function equal to 1 only at 0, actually at 0 there is a problem, if it is 0 then it means that capital omega is 0, capital omega is 0 right, if it is 0 capital omega is 0 so that is a trivial solution. If capital omega is 0 what are we talking about? So what will be the first number, 360 is right, so what will be the, so basically it has to be an integer is not it, because that factor of 2 is there, do not forget. So when k is an integer, positive integer that is when it will become maximum, when will it be minimum minus 1, what is 180, so 2 you are forgetting 2, 2 into 180 is 360. So this nice solution is it is a minimum when this capital omega t plus phi s plus phi equal to k plus half, when it is k plus half multiply that by 2, okay k equal to 1, 3 by 2 multiplied by 2, how much is it 3, so all sin tan cos it will be minus 1, what happens when k equal to 2, 5 by 2 multiplied by 2 it will be 5, so it is minimum there, so what we get is that we see that it is a modulated function with maximum and minimum, we will show you in a moment what that function looks like, but now it is important to remind you once again what E0 is or E0 square is and that is why we are saying so many times that it is a moduloct laser. See the intensity that goes in itself is a function of time, it is not CW, if it is CW then the curve that you get this function will be something like this, it is an oscillation between maximum and minimum, however do not forget that it is not so simple, if you think that we have delta function pulses that is the easiest thing to think, then what will happen, whenever the delta function is there, only in that instant it will be multiplied by the modulation factor and you will get some value, otherwise you would not, alright and we will see how that leads to a decrease in repetition rate, without doing too much of math it is better to show you pictures. If you do not understand this after reading this book, I suggest that you go back and read the original chapter in by EW small in topics in fluorescence spectroscopy volume one like edited by Lakovic or you go back and read the original paper, I will share the original papers, it is interesting to read papers from those era. So the laser that they had discussed was the laser that was used in EW small slab actual spectrophysics synchronously pumped, modulocked, cavity dumped, die laser, volume in 60. So there the repetition rate of the intra cavity pulse is 82 megahertz because it is pumped by 82 megahertz modulocked ND or ND glass laser, okay, so you get pulses like this and we know by now that the time between two pulses is something like 2L by C round trip time, okay. So this is the laser pulse within the cavity not coming out, okay and this is the profile of the sound wave that is applied, radiofrequency that is applied, 779 megahertz, I am sure they wanted to make 800 or 780 but then these numbers worked out to be convenient, okay and this integer plus half timing is a term coined by I think spectrophysics people, where does this come from? Integer plus half, remember what we discussed in the previous slide, maxima at K integer minima at K plus half, integer plus half, so from there it turns out that if you now divide this 779 by 82 megahertz, what do you get, 9.5 is too intelligent a guess, it is, you get 9.5, so what is the integer here, 9, 9 plus half, right, so what happens what this means is again without going into much of math is that every ninth pulse goes out, the remaining 8 go back into the cavity, why does this happen? It happens because you have these modulated pulses, right, so we have this omega plus, omega minus capital omega, some people write omega 0 plus minus capital omega, so these are different frequencies, so they are going to also get superimposed with each other, you will have constructive interference, destructive interference, so on and so forth, right, so that is actually the reason why it happens due to modulation, now if this is the pulse, let me show you a figure of the modulating factor, remember the modulating factor, what is the modulating factor? Modulating factor of intensity, you saw the expression for intensity of output, there was a factor of modulation, what was that, eta, we said that it is periodic looking at it, to remind you we will go back, 1 plus cos 2 omega t, remember that is the modulation factor, so let me now plot the modulation factor, this here is the modulating function and remember modulating function is multiplied by the intracavity laser intensity to give you the intensity of the output, here we are not showing 9 pulses, we are only showing 3 pulses, look at the first one, this blue one is the intracavity pulse and this is modulating factor, you get the final intensity by multiplication of this factor and intensity of this pulse, what will be the product V for this pulse, what is the value of modulating function here, 0, so anything multiplied by 0 is 0, what about the third one, 0, what about this, that is the maximum, so what will happen is, when you multiply the intensity of the moduloc pulses, intracavity pulses by the modulating factor, these two vanish, only the middle one survive and gets intensified, that is how repetition rate is cut down, so that is what we were saying all this time, do not think that E0 is just a constant, it is not because you are not dealing with a CW laser, let us think for a moment that we have a CW, we do not have, we have this laser with 2 high reflectors on the 2 end, gain medium, but no modulocking, it is passed by a CW laser, it is pumped by a CW laser, then the laser output would be something like this, you would get a modulated output, but even then it go down to 0, however here the intracavity laser itself is pulsed, that is why first of all you get pulsed output and you get a narrow pulse, secondly your repetition rate goes down, how much the repetition rate will go down is determined by what kind of capital omega you use and timing and well sharpness of pulse, intensity of pulse, all that is all that can be controlled by controlling the phase, relative phases, end of the day it comes down to turning some knobs for the user, but this is what goes on and so this is how one can produce a cavity dumped output of an already modulocked, cavity dumped pulse output of an already modulocked laser. Now we will go on to something else and that is something we use in our lab of pulse picker, operation of pulse picker is very similar to what we have discussed so far with one exception, in a laser where is the modulocker, in a modulocked, actively modulocked laser or even passively modulocked laser, where is the modulocker, inside the cavity exactly, what about the pulse picker, where is the pulse picker, it is outside the cavity that is a very major difference, so here you use a bragg cell like what you use in cavity dumper, but you use an extra cavity bragg cell, what is the problem with that, the problem is you generate two beams, now you decide which one you want, the one that has undergone diffraction that is the one that will be modulated, that is where the repetition rate will go down, you choose it, but you do not take the other one that is all that part is actually dumped, it is outside the cavity, there is no chance of it to go back into the cavity and get further amplified, so you have a 80 megahertz laser, use a pulse picker to decrease the repetition rate to 8 megahertz, what you do essentially is that you throw away 90% of the energy, it is as we have said earlier a wasteful technique, but unfortunately there is no other option, if you want to work the femtosecond pulses, till date it has not been possible to cavity dump trisaphyl lasers, this is how a pulse picker works roughly, so once again you play around with the phase and all, you get the desired output, that is one thing, let us now move on from the REM of ultra short pulses to ultra long pulses, but there is plenty of use of that as well and when we finish the discussion we will see that the lesson we learn from dealing with ultra long well should not say ultra long longer than ultra short pulses is useful in ultra short laser pulse technology as well, so well we change this, so you do not know what I mean by sequence of events, what we want to talk about now is how Q-switching takes place, Q-switching is a technique by which one cannot produce femtosecond pulses, one can produce picosecond and nanosecond pulses, but these are called giant pulses, pulses that have a lot of energy and in fact you could do something else, as we are going to come back to this again, you could use a Q-switch to produce giant pulses which contain mode locked pulses, so you can produce a pulse envelope within which many ultra short pulses are there, we will come to that later, that part is not very difficult to understand once we have understood the rest. So, the essence of Q-switching is that first of all you maintain the laser in a very low Q level, does somebody remember what the meaning of Q is in this context, we are not talking about Q in James Bond movies, Q is for quality factor and what is quality factor is the ratio of gain to loss multiplied by 2 pi, what is the meaning of gain, what is the meaning of loss, gain means how much energy has been built up in the system, loss means how much energy has exited the system. So, to start with the way Q-switching works, the philosophy of Q-switching is that first of all you have a gain medium, but if you do not have a cavity, right now you consider a hypothetical situation where you can magically create a cavity whenever you want. So, let us say we only have a gain medium and it is being pumped slowly, you have turned the pump on, pump is becoming stronger and stronger initially what will happen, population of the lower energy level is much more than population of the higher energy level, so there will be absorption, there will be loss and all, so to start with you have a high cavity loss, then as you pump, gain increases at a point when gain is nearly equal to loss that is when you switch the cavity on, switch the cavity on means you have gain medium here, you are pumping, to start with let us say I have blocked the mirrors, when I switch the cavity on means I have removed my hands or whatever the shutter is and so now the mirrors are in place, what will happen when I switch the cavity on in this case, now stimulated emission will go back and forth in the cavity and there will be a huge increase of gain until there is saturation, right, so that is what happens and the moment that happens loss will go down significantly because stimulated emission, right, so what will happen is the population of the higher level will keep on increasing, finally it will saturate and then you will have a steady state that you will reach, okay, so this is Q switching, you switch from a low Q cavity to a high Q cavity but then when there is a lot of gain, we are talking about a regular cavity here, regular cavity means a high reflector and an output coupler, okay, so as gain increases and saturates what will happen, eventually a threshold, leasing threshold will be crossed, right and light will come out through the output coupler, then what will happen, then all of a sudden the moment the light comes out, gain will go down again and then you cannot have any more leasing, right, so this is what will happen, this is the sequence of events, start with the loss, high loss cavity then increase gain when it is nearly equal to loss, switch on the cavity so that all of a sudden the loss goes down and then soon afterwards you reach a threshold saturation after which light comes out, the moment light comes out, gain falls and this is when you see now this is from this point on, x axis is time here, from this point onwards the build up of pulse starts and this is what you get, we do have an animation a little later, this is why it is called a giant pulse, whatever energy has been built up, that is given out very quickly, now we are not going to discuss every detail point about this, there are conditions involving excited state, lifetime, what kind of cavity, what kind of gain, what kind of loss, I encourage you to read it from this book, for once we are citing another book, Laser Fundamentals by Silphast, quite an old, actually the book was published I think in 1998 and then they have reprints, quite a nice book but you will appreciate it if you are familiar with a little bit of mathematics and you like optics, if you do not want to do that, this discussion should serve for a qualitative understanding, have you understood, is there a question, can I move on, the question is how do you do it, how do you at your will switch on the cavity, you do it by using something called a Pockel cell, this colourful diagram is from I think Olympus website but you would find more well, informative sources in many places, to put it very simply a Pockel cell is a non-linear optical medium, where here you have a window through which light enters and then in a direction perpendicular to the propagation of light, you can apply an electric field, okay, there is something called Pockel's effect, we will not discuss it here, it will suffice if you know the end result, what it means is that when you apply this medium, it is a property of the medium with which you make Pockel cell, it is a non-linear optical medium, so when a voltage is applied in a perpendicular direction to the propagation of light, if the light is polarized light, application of voltage causes rotation of polarization, okay, and now by how much rotation, so if there is no voltage, there is no rotation, if voltage is there, then rotation is there, so you can for chemists, you can think that this is something like a electro chiral medium, you know chiral media, chiral molecules rotate the plane of polarization of light, here also the plane of polarization of light is rotated but only when you apply an electric field, okay, now the question is by how much does it rotate, does it rotate by 5 degree or 50 degree, 90 degree or 180 degree, it depends on what kind of voltage you apply for how long, okay, so what is shown here is how to produce circularly polarized light, you apply the electric field in such a way that the output is half horizontally polarized light, half vertically polarized light with a certain phase difference, that is what gives you circularly polarized light, if they are not exactly in the same ratio, you get electrically polarized light, if you can convert everything, then you get a 90 degree rotated plane polarized light, half of that you can rotate by 45 degrees or whatever you want, that completely depends on the medium and the voltage that you use, right, and then you can apply the voltage for a small amount of time, tens of nanosecond or microsecond or something, so suppose you applied it for 10 nanosecond, then what will happen, polarization of the light going through will rotate for this 10 nanosecond only, the moment you switch the voltage off, polarization goes back to its original position, right, so this is how a Pockel cell works.