 In last lecture we learned about the weak focusing we derived an equation of motion and this equation of motion was for the more particles motion when there is a dipolar magnetic field having some gradient in it and this combination of dipolar magnetic field and gradient leads to weak focusing. Now there are another and also we learned about the geometric focusing in which shape of the magnet which is known as sector dipole magnet due to virtue of its geometry it focuses in the horizontal plane. There are another types of focusing also available for the charge particle motion and those are strong focusing and age focusing and on the basis of strong focusing there is a principle namely alternate gradient principle in short age principle which is used to make charge particle optics in synchrotron and transverse lens so we will learn these things in present lecture. Now here we recall the focusing of light using the thin convex lens so just I draw convex lens which you have learned during your school courses that this is a convex lens and this is the optic axis and light ray when passes through this lens it bends the light rays to have a focus on a point this is the focal point so it has some bending here theta by which light ray bends during the passage of the convex lens so we also have to have such kind of bending for charge particles if we want to focus these charge particle fields so how this can be achieved we will see this now in charge particle definitely the control of trajectories is using the magnetic field and in magnetic field if charge particle passes the radius of curvature is this we have seen this formula again and again that radius of curvature is rapidly proportional to p and inversely proportional to magnetic field b now if there is a charge particle and if it is passing through a region where magnetic field b is applied then we can calculate what is the bending and that bending you can say that this is the bending by d theta and this is the radius of curvature r so d theta will be q by p dl this is just this is just dl by r and r is placed from here so in this case the total bending of the charge particle trajectory will be q by p and integration of b dl so this is the angle made by a trajectory under the magnetic field of the given length it is the magnetic field which we use for bending the particle trajectory and this is the length over which magnetic field is applied length over which magnetic field is applied b is applied so you can see here that alone b is not enough to decide the charge particle trajectories or angle in the charge particle trajectory rather than the combination of the magnetic field and over the length on which it is applied that is much more important means combination of magnetic field and length it is useful rather than only b so now we see the basic mechanism of focusing how we can achieve the focusing so suppose this is the optic axis in red color and a particle is coming on this trajectory and this is the magnetic element which we want to use as a lens magnetic element when it passes through this magnetic element the magnetic field and length of this element generates a bending theta 1 in the trajectory and trajectory passes through this point here the deviation in the particle trajectory from the design trajectory is x1 so if this happens for all the trajectories then we say this magnetic element is capable of focusing the charge particle rays so we draw another trajectory also which has larger deviation x2 than the previous trajectory then it has larger bending angle to reach the same focus point if we draw another trajectory having opposite displacement means displacement in the opposite direction to the design axis then again it has the opposite angle due to sign in the sign means if we say that sign of these angles are minus then sign of this angle is plus so now we can see by these trajectories that theta is directly proportional to x as x increases the required theta is also increases to send the trajectories on the focus point and if x changes its sign theta should also change its sign therefore now we have seen in the previous slide that theta is directly proportional to bdl d theta is directly proportional to bdl means this theta means bdl it should be directly proportional to x then focusing can occur so if this combination we are able to generate proportional to x then that kind of magnetic field will focus the charge particle in geometric focusing b was fixed because that is the dipole magnet so that was constant over the space however the length which the trajectory passes through the magnetic field has different for different initial displacement and then different dl or l needs to focusing and in the magnet dipole magnet if we introduce the gradient then bending as well as focusing is achieved that is known as weak focusing and in that case field index must be between the 0 and 4 for stable motion in both the planes can we separate the bending and focusing can we make a magnet which is having only gradient and no dipolar component in that case no bending of the design path will be there because dipolar magnetic field is absent and there is only gradient so focusing can occur this type of magnet will behave exactly the length as of the light optics so answer is yes we can do that if we can generate such kind of magnetic field in which vertical component of the magnetic field increases with displacement x from the design axis and the horizontal component of the magnetic field increases with the vertical displacement then this kind of magnet can be used exactly similar to the optic convex lens in the optics or concave lens in the optics now you can see here on the design axis we have x is equal to 0 and y is equal to 0 these are the coordinates on the design axis because on design axis itself defines the origin so x is equal to 0 y is equal to 0 these gives p x is equal to 0 and p y is equal to 0 for this case and p x is equal to 0 and p y is equal to 0 means there is no magnetic field on the design path means design path traverses this magnet without getting affected there is no effect of this magnet on the design trajectory and this is the case similar to the light optics that optic axis does not change when it passes through the convex lens or concave lens and as x becomes non-zero or y becomes non-zero there is finite magnetic field and this magnetic field is used to bend the particle trajectories which are deviated from the design axis and this angle of the deviated trajectory leads to the focusing this kind of magnet is known as a quadrupole magnet because this field can be generated if we have four poles rather than two poles now how these four poles will be erased we will see it now by by as a gx which was written so this is the graph showing the variation of by with x this is a simple straight line having some inclination with x axis and this inclination defined the gradient what is the gradient g now you can see that as x increases here x is increasing here the magnetic field also increases means the applied force by the magnetic field on the charge particle will also increase so as we go further and further from the design axis this is the design axis which is x is equal to 0 y is equal to 0 origin here is the design axis coming out of the screen and if particle is deviated along the x axis the more deviated particle feel more force means more bending and it is the design characteristic of the focusing as we have seen that if x is more theta should be more and if x is negative here is x is negative and here x is positive so in the case of negative because by is also has changed its sign so force will also change the sign and it will be in the diverse direction of that of this direction so now again if particle is deviated towards negative x it will get a force towards the positive x and if particle is deviated in the positive x it will get a force in the negative x so overall all the particles will feel a force which will send those particles towards the design axis it means this is the kind of focusing we are trying to confirm the particles towards the design trajectory now how the four poles are arranged these four poles are arranged in this manner this is a north pole this is a south pole this is again a north pole and this is a south pole these four poles make a quadrupole magnet this is the pole phase profile now like the conductor in the conductor the electric field line always meets a right angle in a ferromagnet surface the magnetic field lines meet at right angle so these are the magnetic field line which emerges from the north pole to south pole and due to symmetry here it is the north pole at this distance and similarly here is the north pole at this distance here south pole on the same distance and south pole tip here is also a same distance so due to this symmetry magnetic field will be zero at the center which is the desired characteristic of a lens and as we go away from the design trajectory you can see the gap between the pole faces is reducing and this reduced pole gap generates a higher magnetic field and higher magnetic field means more force so larger deviation in the particle coordinate will give a larger kick by this magnet now you can see here if we take four points say this is the point number one this is point number two and this is point number three and this is point number four so point one and two are having deviation along the x axis while point three and four are having displacement along the y axis now at this point the magnetic field line is in just this direction that is the tangent at this location on the magnetic field lines here the tangent is in this direction so magnetic field is in this direction you can see that when x is positive magnetic field is in the opposite direction when with the case of when x is negative the similar is the case for y axis that on the positive y axis and negative y axis the magnetic fields are in the opposite direction now if a particle comes here suppose and particle comes here on the point three and point four you can see if particle is coming it is coming out of the screen and magnetic field is in this direction so force will be v cross v so v is outside the screen v is in this direction so force will be in this direction similarly here because the magnetic field has the reverse direction then this place so force will also have reverse sign in the direction so force will be in this direction so force on a positively charged particle which is coming outside will be in this direction on the vertical axis so force is towards the design axis now suppose the division in the horizontal axis now you can see if you are taking the positive charged particle which is coming outside similar to the case of point three and four you will see that the direction of force will be away from the axis so this is not desired here means this type of quadrupole magnet is focusing in the vertical plane why it is defocusing in the horizontal but we have to live with it actually Maxwell's equations compels us to have dy is equal to gx and bx is equal to gy and due to this we have focusing either in horizontal plane or in vertical plane in the other plane there will be always defocusing so suppose if you swap the poles means north is swept by the south and south is swept by the north means we have rotated the quadrupole magnet by 90 degree then the focusing will occur in the horizontal plane and defocusing in the vertical plane means if we consider a single plane and we place two quadruples just rotated by 90 degree then first quadrupole if it is focusing then second quadrupole will be defocusing means we have a convex and concave lens combination similarly will be the case for the other plane here concave and convex combination will be here so using these two combination we can always make focusing in both the planes now what is the focal length of this lens this is the particle trajectory which is coming when it passes through this magnetic field it generates a bending angle theta and particle trajectory crosses from this point so for obtaining the focal length we have to take initial condition in which x prime is zero means this trajectory is making zero angle with respect to design axis means it is coming parallel to the design axis and it has certain deviation x and this is from where it cuts or it crosses the design path it is the focal length so due to simple trigonometry we can obtain the time is equal to minus x by 10 theta here I am taking this sign as a minus and in paraxial approximation where theta is small theta is small we can always have can have 10 theta is approximately here and this approximation is again used here so our focal length is x by theta x now from previous slides we can put the value of theta x in terms of b and l now here theta x will be q by p integration of b by dl and by for the quadrupole magnet as gx so it plays of by we can put gx so this will be gx dl and g it is the gradient so gradient will be constant for a given quadrupole magnet we have seen that in our this slide that this gradient is constant for a given quadrupole magnet because this is a straight line so gradient will not change only magnetic field will change so gradient is constant in the quadrupole magnet so this gradient is constant we can take it outside the integration so this integration will be q by p g integration x dl now if we take very very small length for the quadrupole magnet means g l is very small means we are talking about the thin lens approximation means dl tends to zero this is the thin lens approximation in thin lens approximation we can assume that x remains constant inside the magnet just you can see that this is a delta function where the magnetic field appears so just before the magnet and after the magnet x doesn't change only there is a change in the x prime which was zero earlier before the magnet it changed suddenly after the magnet and has theta x minus theta x in this case so x can also be taken outside the integration so this integration will become q by p gx integration dl and dl is just the integrated length of the quadrupole so this will be q g by p xl and this quantity gradient normalized by the momentum and charge is known as normalized quadrupole strength k so quadrupole's bending is equal to kxl we see this is a quadrupolar kick when kick is proportional to x here we can say theta x is proportional to x l and k are constant for a given quadrupole so as x changes theta also changes so theta is directly proportional to x in case of dipole magnet this theta was independent of x so this kind of kick is known as quadrupolar kick and by comparing this two equation this equation and this equation we can see that focal length is just minus one by k i minus focal length is for focusing for a diverging lens or concave type lens this will be positive so you can see a equation here trajectory here after a defocusing quadrupole this trajectory goes away from the design axis and in this case theta x is positive this is positive that's why focal length is also defined as positive in the diverging lens now suppose we want focusing in both the planes simultaneously one quadrupole it is focusing in one plane then it will be focusing in the another plane so for overall focusing in both the planes we at least require two quadrupole magnet so in any optics of charged particles you will always see that at least there are two quadrupole magnets in two quadrupole magnets in one plane if it is focusing and defocusing combination so for the other plane it will be defocusing and focusing combination so by properly choosing the strength of these two quadrupole magnets and the distance between these two quadrupole magnets you can see that these four trajectories can be focused means in both the planes we can get focusing and we may recall your lens making formula so this is the overall focal length of two lenses combination that is one by f1 one by f2 minus d by f1 f2 f1 is the focal length of the first lens f2 sorry it is it should be f2 is the focal length of second lens and d is the distance between these lenses so this is also true in this case also so you can always calculate the focal length of the combination of two quadrupole using this problem remember this is valid only when we are using thin lens approximation if quadrupole is not thin then we have to calculate explicitly what should be the x and x prime at the exit of the quadrupole magnet and what will be the effective focal length of that combination we will learn these things when the quadrupole is not thin in the next lecture