 going to discuss lecture 7, where we will be discussing a very important aspect of accelerators that is beam focusing. And this beam focusing is required for all the experiments and you can see that the beam extracted from iron source or even accelerated from an accelerating tube normally is diverging. For example, if you take the iron source here, there is a plasma here and there is an extraction voltage, then the beam will be going almost like this because beam is coming from all places. So, it will be a very diverging beam. Now, this beam has to be focused before it is taken to the target. Otherwise, what will happen if the beam size is big and target is small, then the entire beam will not be used and the results will not be accurate. Therefore, it is absolutely necessary that before taking the beam accelerated beam to the target, it has to be properly focused and it will always be preferred if the beam is parallel where it is going to the target. If it is a parallel beam then it will be preferred. This has been done in the case of, for example, light beams. It is done with the glass lenses and in the case of electrostatic lenses, iron beam which are accelerated, electrostatic lenses or the magnetic lenses are normally used for getting the beam focused. And normally the target size is not more than the few millimeters to one centimeter and therefore beam has to be focused to that. Then you will see that both electrostatic and magnetic lenses are used and in fact, even the dipole magnet which we always think that dipole magnets are used only for deflecting the beam. If they are given, their pole faces are given the proper shape in order to get the magnetic field components energized then they also can be used for focusing the beam. But otherwise normally electrostatic lenses either electrostatic quarter pole or maybe angel lens and the magnetic quarter pole lenses are very popular for focusing the beam. Then you will see that if you are using these doublets that means two quarter poles either electrostatic or magnetic. Then you will see that one quarter pole always focuses in one plane and it defocuses in the other plane and therefore if you want to have the focusing in both the planes then you have to use doublets, minimum doublets or if you want to have the symmetry of the beam, total rotational symmetry then it is required to use a triplet rather than the doublet. At these details you will see that these quarter poles are useful for both low as well as high energies. However, for very low energies another simple lens which is called angel lens which requires only one voltage and other two are grounded normally but it is not necessary that they should be grounded and they can be used. So there are advantages and disadvantages. If you use magnetic quarter poles then you will find that this focusing is mass dependent or the velocity dependent. However, in the case of electrostatic quarter poles this the magnetic field or the electrostatic field strength does not depend on velocity and it only depends on the charge and therefore they are much more convenient to use. However, at low energies always electrostatic quarter poles are preferred because they are supposed to be stronger than the magnetic quarter poles and of course the velocity independent therefore more convenient to use it. At high energies these electrostatic quarter poles cannot be used because the voltage which is required to focus these beams has to be of the order of the energy. So suppose you want to focus the some 50 keV beam then the electrode will require voltage of the order of the keV beam and therefore at high energies the magnetic quarter poles are preferred and they are much better with those limitations of mass dependent and therefore the current which has to pass to generate the magnetic field has to keep varying with the mass and the velocity. You can see here there is a beautiful analogy between the optical lenses and the electrostatic lenses you can see here you can see the electrostatic quarter pole. All these electrostatic or magnetic quarter poles they have four poles and for example in electrostatic it is called positive positive and negative negative so there are four poles and these are these are equiptual lines. So you apply the voltage here let us say it is V here and any voltage either it can be grounded or it can be more or less there. Now if you see the analogy then let us take a glass lens which is converging type here and all the beams all the particles here at different distances from the optical axis at the distance of D at various D's will be focused here at the same point because the curvature changes so the beam which is coming very close to the axis will go almost like parallel but the beam which is coming very or hitting the lens at larger D's will have much larger deflection angles. So it is that theta depends on the D the distance from the optical axis to the place where beam is hitting the lens so you can write if you write this then it is almost like you can say the tangent of this theta is equal to D by F if F is the focal length which is here to here this is focal length and so you can say that the and if the theta is small then you can write in a simple expression it is D by F for small D's. So if the farther is the particle from the axis the stronger is the focusing force and therefore it depends linearly for the small distance D from the optical axis. Now in electrostatic lenses there are two types as I mentioned one is called angel lens and other four the other one is called water pole lens electrostatic water pole lens and in a quarter pole field configuration for this one is shown here you can see that the particle will be focused properly if the opposite poles are of the similar quality. Now in the case of electrostatic water poles which is shown here that this is plus V here both and this is minus V the field which is a function of an x and y let's say that this is x this is y axis and this is x axis and here the z axis is normal to this which is a beam direction in that case with following geometry the field or the voltage distribution which is a function of x and y depends on k y2 into x square minus y square this of course is exact if the geometry of the poles is hyperbola and then this equation will be used and of course I am assuming here that beam is along z axis the field errors of course has to be minimized and they depend on two factors mainly one is the manufacturing and that is the accuracy and the other one is the positioning accuracy when you are aligning the things so if there is a error in the either manufacturing or the positioning of the quarter pole then there will be errors and which will disturb the ion trajectories as well as the focusing effects of the quarter pole these errors not only can change the focusing effects but also can change the motion of the ions and they can induce the aberrations which I have mentioned this is true for both electrostatic as well as magnetic quarter poles I have shown here the first one is angel lens you can see here this is angel lens and angel lens has three electrodes and normally two electrodes which is at the entrance at the exit they are normally grounded they are part of the beam line and the center electrode is given the voltage but it is not necessary that these other two should be grounded they can be isolated also and if you isolate then the voltage is V1 and V1 in the beginning first electrode and the V2 at the last electrode and the central electrode will have this however the voltage difference is V-V1 and V-V2 in the case of you can see the field lines which are responsible for the focusing of this so basically this angel lens also is a three electrode or they can be three cylinders and these cylinders are applied voltage and as I mentioned that normally these two electrodes one is at the entrance and the other is at the exit they are grounded so if suppose the beam line is coming here then only central electrode is isolated and then these two are grounded they become the part of the beam line so they are grounded and therefore you don't have to apply voltage so this V1 and V2 becomes 0 so V1 is equal to V2 and is equal to 0 means it is grounded it is a part of the tube or part of the beam line and only this is applied the voltage V you can see that how this works you can see the field lines here and at any point for example here there will be a force in this direction and this will have two components one in this direction and other one one is this one with other one is this and similarly here there are two components and they will be responsible for focusing the beam you can see here that if one is focusing the other one will be defocusing because the field direction will be reversed and therefore if first one is focusing or defocusing in this case then this will be focusing so it will work like this but the total focal length will be positive that will be the next effect of this lens will be the focusing now coming back to so this is a very simple but only problem is that this V the voltage which you have to even if V1 is equal to V2 is equal to 0 means they are grounded you have to apply voltage on central electrode which is V and that voltage has to be pretty high if you want to go to higher energies and therefore these lenses are mainly used for lower energies but they are very accurate, very simple, very cost effective and they are in general used for low energy accelerators and very effectively focus the beam however this is the quadrupole lens electrostatic lens and this again like magnetic lens which I will explain later will focus in one plane and defocus in another one so they always have to be used in doublet and minimum doublet and if you want a cylindrical symmetry then of the beam then it has to be you have to have a symmetrical triplet now in this case you will see that if first quadrupole is focusing in X plane and defocus in Y plane then the second quadrupole which is rotated by 90 degree will defocus in X plane and focus in Y plane so that effective focal length will be positive and it will be effect will be net effect will be focusing the beam so this you can see here that as I mentioned earlier that both electrostatic and magnetic quadruples they have four poles here it was positive, positive, negative, negative here it is north, north, south, south so you can see that if a particle is going particle in all the calculations particle direction is always taken as Z as I mentioned let's say this is Z this is a beam axis Z is always beam axis and these are called transverse axis this is called Y axis and this is X axis which is perpendicular so if Y is like this X is like this and Z is like this so these are three axis which you can see and at any point the magnetic field which is between north pole and south pole so magnetic field will always be like this and from here to here and this place to this place here for example here so you can see that if a particle beam is slightly defocused and if a particle is falling on this point P then you will find that it will face it will see the magnetic field B is perpendicular to this and it will have two components B, Y and B, X which are responsible for focusing and defocusing so you will see if you can analyze which you will see in the next slide that it focuses in one plane and defocuses in another plane and for doing that you will see with the analysis that unless you use the quarter pole doublet you cannot get the focusing in both the planes and therefore they are always used these quarter poles are always used either as a doublet or as a triplet you can see I am giving the summary of this that you saw that since the profile of the pole is hyperbolic in ideal case then as we move along this axis the magnetic field increases which is B, Y here this is B, Y and this is B, X and so as X increases the distance this keeps coming down this distance keep coming down and therefore the B, Y will keep so we can always write that B, Y is proportional to X here and therefore you can write it is equal to G times X this G is a constant of proportionality it is nothing but a gradient and similarly if you go to Y axis side then also you can write that B, X is proportional to Y here as you go and it is also again like a G into Y this G is proportionality constant here now the force on this particle coming here at this point will be written as Fx you can see here that how this total force is both is E charge times E plus V cross B and B I said that B is always along the Z axis so VZ is really equal to V now in this case when we are focusing with the quadrupole magnetic quadrupole E is 0 so this point is 0 this point is 0 so this will not appear and therefore the force will be force on the particle which will be responsible for focusing or de-focusing is nothing but charge time B cross it is a cross product of velocity and the magnetic field now these are all vector quantities so you can write this force as I Fx plus J Fy plus K into Fz so these are three components and the right hand side you can write in the matrix form and if you expand the matrix form here then you can write this equation and from this equation you can write Fx is equal to charge times By Vz By this is velocity and this is magnetic field as I mentioned that we are assuming that velocity is along the Z axis so this will be 0 this is 0 this is V this will be V so you can write that this is nothing but Vx is equal to minus charge times Vz Z component which is as I said that if the velocity is in this direction this will be equal to V V is the velocity into the Y component of the magnetic field and you can write similarly equation from this you can write that it is By is there now you can see here if you compare Fx is minus this and Fy is plus this so you can see that in one case if it is focusing in X plane it will defocus in Y plane and if it is focusing in Y plane then it will defocus in plane and therefore it is necessary that by proper choosing the parameters you can focus the beam in both the planes by rotating it because you have to change the direction you have to change the sign of that so you have minimum two quarter poles one here and one here and this quarter pole is rotated by 90 degree so that the field directions will change and then it will give the focusing in other plane where it was defocusing in the first quarter pole now ultimately having known this so this has to be calculated properly and of course in the quarter pole not only you have to use two quarter poles if you are using magnetic quarter poles they can be closed by between two quarter poles there has to be certain distance so you have to consider that also there will be focusing if the first one is focusing in one plane then it will keep focusing if there will be a drift tube like of thing you have to consider that also so ultimately what exactly we want we want that we want to see how much is the focusing or defocusing and for that we have to do the we have to study the motion of the charge particle both in X and Y plane X and Y plane are transferred plane and this velocity is going in longitude level so we have to see the direction motion of the particles in the field whether it is electrostatic or quarter pole magnetic in both the X and Y plane so let us and they will be somewhat similar so you can say that if I study thoroughly the motion of the charge particle in X plane it will be somewhat similar to that in Y plane also and then it will change when in the other quarter pole which is 90 degree rotated so you can write that if you want to study the motion of the particle in X plane then you will see that the force is Fx which is responsible and force is nothing but the derivative of momentum that is dp by dt and this is nothing but d by dt of mv which is the momentum and therefore you can write this as d by dt within bracket m into dx by dt because velocity in X plane is dx by dt and this you can write as m into d square X upon dt square this is a this is called acceleration this is nothing but X double dot derivative of velocity so if you write all these things and the force coming from this equation here these equations here then you will find that our equation becomes like this this is the equation you take it to right left hand side then this becomes the equation so this is the equation which we have to solve in X plane X plane so suppose you write this as let's say k and change the word see ultimately you want to know the trajectories as a function of z because it is the velocity of particle is moving in z direction so you don't know time of course you can calculate velocity but that is not important so what you want to know X as a function of z and Y as a function of z these are the transverse planes so you want to calculate the trajectories of particles in X plane X plane and which is as a function of z in the X and Y plane both of them so you have to convert you have to change the variables from t to z here and you can write that z is nothing but velocity time the difference and that if you differentiate it d z is equal to v z into d t and v z is nothing but velocity is velocity we are taking the particle is moving only in the z direction so if you write this then the equation becomes this is the equation which you have to solve where if you write k X as this e type g upon m v z then this becomes the equation which you have to solve and this is a very simple equation which will have a solution where X which is a function of z you can see this is written here it is a very simple one and if you differentiate it then X prime becomes like this this is the X prime so these are the two equations you have to solve first is that how the x is going to be as a function of z and how the divergence is increasing or decreasing so you have to solve these equations simultaneously both of them and see them now here you can see that in this equation these are they are two constants a and b and they have to be formed from the boundary conditions and there so effectively when we say focusing the solution of these two equations and find out that how the trajectory is moving along z so these are the two equations which have now if you do that in a very simple terms and this one by f is given here and which depends on g and where l is the length see for example if you have a quarter pole here let's say you have quarter pole here normally these quarter pole lenses magnetic quarter pole lenses they have a finite distance let's say finite length here so this l is a length while you have seen in the case of optic lenses glass lenses they are also having two types of lens one is thin lens and thick lens so in the case of thin lens you can neglect the thickness suppose it is let's say in the case of converging lens it is a very thin lens then you can forget about you can omit the length but if they are suppose this length this lens was very thick then you cannot do that see similarly in the case of electrostatic either quarter electrostatic quarter pole or magnetic quarter pole this length cannot be avoided if it is not a thin lens so if l is the thickness of the lens then you can write the focal length here which is focal length here and that is a function of velocity not velocity magnetic field rho which is a radius of curvature length or the thickness of the quarter pole and the g factor which is coming from here you can see that g is coming from here that's a gradient of the magnetic field if you put that simply then you can see here that 1 upon f l is k times this and these are two both are constants and here the k is called the focusing strength and at this moment I would like to again tell you that if you have two lenses it's not that it will give de-focusing beam one is focusing and the other one is de-focusing so suppose you have a combination of two lenses you have a combination of two lenses then if a net effect will be focusing so suppose either you have focusing and de-focusing or you have de-focusing earlier and the de-focusing here like this and the focusing later on is still the effect will be and that you can put still it will be focusing effect only and therefore if the beam is coming here at let's say at let's say 2 cm let's I am just putting it then you will get the beam here focused at certain distance and it will be much smaller it will cross at this so there will be some focal length and that focal length will be equal to fc which is called combined focal length and that is equal to 1 by f1 f1 is a focal length of the first lens plus the 1 by f2 f2 is a focal length of second lens minus the d and d is the distance between these two d is focal d is the distance gap between two lenses so if you put that you can put any one of them either f1 you can put negative that means de-focusing or you can put positive and the other one other way now if you either of them you will find that it is always the combined effect the combined effect is always positive and therefore any combination of lenses either focusing focusing lenses or converging converging or converging diverging or diverging focusing or any combinations you have this fc combined focal length will always be positive and therefore the net effect of this will be this will be net effect will be focusing so this is just to give you an idea and here I have given some idea about the about the how the the mathematics works and how the trajectories can be calculated and similarly you can do in y plane also and if you do this whole thing the trajectory calculation and see that how the focal lens in x plane and y plane looks like then you see that these focal lens are given here x plane and y plane both and you have a focal length here and the trajectory will be when you want to calculate the trajectory then you have to inject a parallel beam here and then see that at what point they will be upward they can be aberration the proper machining of these lenses is not done here it is deflected so here you can see the parallel beam coming here is focusing at this plane and here in this case it is diverging it is a diverging lens so the trajectory is going like this so it will be crossing the optical axis at a point opposite to that so this becomes like minus f so this is direction so that is why I said that and normally in the case if you want to have a good quality beam image then this focusing and de-focusing full focal lens in magnitude are equal they are kept equal the design is done in such a way that they are equal and these are almost identical and therefore you can get see idea of in any experiment idea is that you should have smallest beam possible and as far as possible it should be very close to the circular beam and that you can have circular beam if the instead of double edge use if you use double edge and analyze the whole thing in the same manner using the same philosophy which I explained just now then you will find that with triplet you can get a very symmetrical beam rotational symmetry you will get it and aberrations will be very small