 Hello and welcome to lecture 5 of the second module of this course on Accelerator Physics. Today, we learn about the accelerating structures, we learn about the pillbox cavity and the drift tube liniac. In the last lecture, we studied about the behavior of electromagnetic waves inside waveguides and cavities. In particular, we studied about the rectangular waveguide, the cylindrical waveguide and also the rectangular and cylindrical cavities. So we saw that in a waveguide, the wave forms a standing wave in the direction in which there are conducting boundaries in order to satisfy the boundary condition at these boundaries. In the direction in which there are no boundaries because the waveguide is open in one direction. So in that direction, it is still a travelling wave. In a waveguide, any frequency above the cutoff frequency is propagated. If the frequency of the electromagnetic wave in free space is below the cutoff frequency of the waveguide, then it will be attenuated. The wave propagates in the waveguide in the field pattern of the modes that are allowed. So several modes are allowed. So for example, in the waveguide, we saw rectangular waveguide, we saw the TE10 mode, the TM11 mode. So different modes correspond to different frequencies and different field patterns. So the wave, the electromagnetic wave, when it enters the waveguide, so it will propagate in the field pattern of the modes that are allowed. So in a cavity, since there are conducting boundaries in all the three directions, so the cavity is now closed at P, closed in all the directions. So the electromagnetic wave here is a standing wave in all the three directions. There is no propagating wave and the standing wave is formed in all three directions. So here now, unlike the waveguide where if the frequency of the electromagnetic wave was above the cutoff frequency, the wave could propagate, here only discrete frequencies are allowed in a cavity. So only discrete frequencies, only at discrete frequencies of the modes, the wave will enter the cavity and it will form the standing wave pattern corresponding to that mode. Any other frequencies reflected well. So today we will study about the modes or the field patterns in a pillbox cavity. So a pillbox cavity is simply a simple cylindrical cavity that is made up of conducting material of length L and radius Rc. So it is a simple cylinder, it is closed in all directions and the length is L and the radius of the cavity is Rc. So as you know electromagnetic waves, when they enter inside the cavity, they will set up in TM or TE standing modes in order to satisfy the boundary condition. So TEM mode as we have seen is not possible either in a waveguide or a cavity. Once it enters the cavity, the electromagnetic wave will be either a TM mode or a TE mode. And I also mentioned in the last lecture that in a cavity, the TM or TE here the transverse refers to transverse with respect to the z direction because there is no propagation. So it is transverse now with respect to the z direction. For TM modes, the magnetic field component is perpendicular to the z direction. So you have for the TM mode the magnetic field which is in this direction. So you will have an EZ field, okay. Similarly for TE mode, the electric field component is perpendicular to the z direction and you have EZ is equal to 0. Now in the pillbox cavity, the electric and magnetic fields everywhere, it should satisfy the boundary conditions. The boundary conditions as we have seen, they are the tangential component of the electric field is 0 at the boundary and the normal component of magnetic field is 0 at the boundary. So this means that, so these are the cavity walls and this is the side wall. These are here the side walls of the cavity. So in order to satisfy the boundary condition, EZ is equal to 0 at the cavity walls. Now when you have EZ, so at this cavity wall here, EZ is the tangential component. So it should be equal to 0, okay. Similarly E theta is equal to 0 at the cavity wall. So E theta will again be like this. So again it will be tangential component. So E theta will be tangential component at this boundary. So it should be 0. ER is equal to 0 at the side wall. So if you have ER, it will be in this direction, okay. Now at this boundary, it is the normal component, so it need not be 0. However at this boundary, at the end wall boundary, it is the tangential component. So it should be equal to 0. Similarly for magnetic fields, BZ is equal to 0 at the side wall. So BZ is this component. So for the cavity walls here, it is the tangential component and tangential component of magnetic field need not be 0. The normal component of magnetic field is 0. So however at the end walls, at the two end walls, it is a normal component and it has to be 0. B theta is allowed at all the walls because at the cavity walls, B theta is the tangential component. Also at the end walls, it is the tangential component and by boundary condition it is allowed. BR is equal to 0 at the cavity wall. BR is in this direction. So at this boundary here, it is the normal component and normal component of magnetic field is not allowed. So BR is equal to 0 at the cavity wall. So with this understanding of the boundary conditions in a pillbox cavity, let us see the different modes of the cavity. So as we discussed in the last lecture, for the TM mode, BZ is equal to 0. So EZ will exist. Now EZ if you see in the radial direction, it has a Bessel function dependence whereas in the theta and Z direction, it has a cosine dependence and being a, it is a standing wave. So there is no propagation in any direction, okay. So here the frequency of the mode is given by this formula. So here as we discussed in the last lecture, the x square, xmn is the nth 0 of the Bessel function jm, okay. And now here m, so in the rectangular, in the rectangular coordinates m, n and p, they simply denoted number of half period variations in that direction. However, in the cylindrical coordinates, the definition is slightly different. Here m, which can take values from 0, 1, 2, so on, it is the number of full period variations in theta of the field components. That means you take any field component and you plot it in theta from 0 to 2 pi, okay, any field component E or B. So how many full period variations you have? If there is no variation from 0 to 2 pi, it is 0. If there is one full period variation, it is 1 and so on. n is the number of zeros of the axial field components in the radial direction in the range 0 to RC excluding at, excluding at r is equal to 0. So in the r direction, so it is the number of zeros of the axial field component. So here you have to take EZ, the axial field component is EZ. The axial field component is EZ, it is along the axis. So the number of zeros of the axial field component in the radial direction. So how many times this field goes to 0? Now in the radial direction, the dependence is of the form of Pessel function. So if n is equal to 0, it will be J0 for something like this. This corresponds to n is equal to 1 or if n is equal to 2, it could be something like this and so on. P is the number of half period variations in the Z of the field. So for P, you could take any fields E or B and see its variation in the Z direction from 0 to n. So if there is no variation, P is equal to 0. If there is one full period variation, then P is equal to 1 and so on. Sorry, one half period variation P is equal to 1 and so on. So these are the Bessel functions. So we have seen this already, these are the variation of the Bessel function. So this is J0, this is J1, J2 and so on. And the values of these zeros can be found out from the table. This is the differential of the Bessel function and again these values can be found from the table. Now let us try to understand this again. So M as I have already explained, M is equal to, it is the number of full period variations in theta of the field component. So you take any field component, it could be E, R, E theta, B, R, B theta, any field component and you vary it in theta direction keeping R and Z fixed from 0 to 2 pi. If there is no variation, if the field is constant, then it means that M is equal to 0. If there is one full period variation, then M is equal to 1 and so on. Now N is the number of zeros of the axial field component in the radial direction in the range 0 to Rc, Rc is the radius of the cavity excluding the 0 at R is equal to 0. Now if you see the form of EZ, it is like this and notice that whenever it is a TM mode in the cavity and you have an EZ field, it will always be 0 at, so EZ will always be 0 at R is equal to Rc in order to satisfy the boundary condition. So this and the variation is of the form of Bessel function. So it depends upon the value of M, which Bessel function variation will be there. If M is equal to 0, then it is J0, if M is equal to 1, then it is J1. So here now you can see here that this is the variation in the R direction. So here M is equal to 0, so we are seeing the variation of J0 and here it is going to 0 once at R is equal to Rc. So this denotes N is equal to 1. Similarly here now if J0, so here again M is equal to 0 and so we see the variation of J0 and J0 goes to 0 2 times or in other words EZ goes to 0 2 times in from 0 to Rc. So here N is equal to 2. Now here in this case M is equal to 1, so we see the variation of J1. So J1 goes to 0 twice, but by definition we should not count the 0 at R is equal to 0. So this is still N is equal to 1, it is the number of half period variations in the Z of the field. So you take again any field component and you see the variation in the Z direction from 0 to L, if there is no variation it is P is equal to 0, if there is one half period variation P is equal to 1 and so on. So for seeing the values of M and P we can use any field components, however for seeing the value of N we have to see the EZ component of the field only. Here are some of the fields, some of the modes in a cylindrical cavity, some of the TM modes. So the most fundamental mode in a pillbox cavity is the TM010 mode. So here M is equal to 0, N is equal to 1 and P is equal to 0. So this figure shows the magnetic field and this figure shows the electric field. So M is equal to 0. So if we see, if at some fixed R, let's say at this R and this Z, if we see the variation of the magnetic field, so there is this B theta component here which is shown here. So we see that there is no variation in going from 0 to 2 pi. So if you plot B theta with theta from 0 to 2 pi, there is no variation in the field. So here M is equal to 0. Now N is equal to 1 and also M is equal to 0. So we have to see the variation of J0 for EZ with R. So we plot the variation of EZ with R. So J0 at R is equal to 0 is maximum here and then the field falls to 0. It has to go to 0 at R is equal to R0. So you can see the fields here, the electric field here. It is maximum at R is equal to 0 and then it goes to 0 at R is equal to R0. So N takes the value 1 here, P is equal to 0. That means there is no variation in the field in the Z direction. So if I see the EZ field along the Z direction, I see that at fixed R and theta, there is no variation in the field. So EZ is constant from 0 to L. So that is why M is equal to 0, N is equal to 1, P is equal to 0. Now the next mode is TM011 mode. So here now P is equal to 1, the other fields remain the same, P is equal to 1. So as again, if you see the variation of the field in the theta direction B theta, there is no variation. So M is equal to 0. If you see the variation of the field at a fixed Z and theta, so if you see the variation of EZ with R, okay, so you will see that it is maximum at the center and then it goes to 0 at R is equal to RC. So N is equal to 1 and now P is equal to 1. So you have one half period variation in the field in the Z direction. So EZ, if you see EZ here, it is in one direction, okay, in between EZ goes to 0. See there is no component of EZ here, this is the radial component at the center. So there is this EZ component here, it goes to 0 here and then it is in the opposite direction. See here it is maximum in one direction, goes to 0 at the center and then maximum in the opposite direction. Now the next mode is TM110 mode. So now here M is equal to 1. So that means there is one full period variation in the field in the theta direction. So if you see the theta direction here, so if you plot B theta, because it is the most convenient thing to plot here, so let's start from here, B theta is 0 here. So you can see here there is just the R radial component here, okay, it goes to maximum somewhere in the center. Now see this is all in the direction, theta direction here, the field here and then again it goes to 0 here and then again maximum in the opposite direction and comes back to 0. So if this is 0.1, this is 0.2, this is 0.3 and 4. So this corresponds to 1, 2, 3, 4 and coming back to 1. So you have one full period variations in the field in the theta direction, okay. Now here M is equal to 1 and M is equal to 1. So we have to plot J1 versus R, EZ will have a variation of the form of J1 here because M is equal to 1. So J1 if you see from the graph it is 0 at the center and then it goes to 0 here at R is equal to RZ. So again this corresponds to N is equal to 1 because the 0 at R is equal to 0 is not counted. So if you see the field EZ with R, okay, so if you see the field, so here it is 0, you can see here it is 0 and then it goes to maximum and then again to 0 here because it has to satisfy the boundary condition. So this is how the field looks like. Again if you see the EZ field along Z there is no variation. So this corresponds to P is equal to 0, okay. Now the next mode is TM111 mode. So it is similar to the mode above except that now P is equal to 1. So now you have one half period variation in the field in the Z direction which you can see from here. Let's take it at this. Let's see the variation of EZ. Let's see the variation of EZ with Z at this point at this R and Theta. So we see that the field is in one direction, EZ it goes to 0 here and then it is in the opposite direction. So it is like this. Now similarly if you have a TM210 mode. So now M is equal to 2 here. So you have two full period variations of the field in the Theta direction. So you see this is the Theta direction. So we can start from here. This corresponds to if you plot B Theta, this corresponds to 0 value of B Theta. So here the field is all radial and then it goes to a maximum here again 0 here. Again goes to a maximum in the opposite direction and goes to 0 here. Then again changes sign and goes to a maximum here and then 0 here and then again another maximum and finally to a 0. So you have two full period variations of the field in the Theta direction for a fixed RNZ. Okay and now if you see the variation of EZ. So here N is equal to 1 and M is equal to 2. So you have to see the variation of J2, you have to see the variation of the Bessel function J2. J2 is again 0 at the center and then increases like this and then goes to 0 here. So again it goes to 0 at this point. This corresponds to N is equal to 1 because this 0 is not counted. P is equal to 0. So there is no variation in the field in the Z direction. The next mode is TM012. In this mode again there is if you plot B Theta with Theta. So if you plot B Theta with Theta there is no variation in the field. So it is constant N is equal to 0. Now since M is equal to 0, you see the variation of J0 for EZ. So here N is equal to 1. So the first 0, so you see that the field is maximum at the center. EZ field is maximum at the center and then going to 0 here. P is equal to 2. So now we have two half period variations of the field in the Z direction. So you can see from here the field is maximum here in one direction, here it is 0 because there is no EZ component, there is only ER component. Again it is going to maximum in the opposite direction, going to 0 here and then again changing sign and become maximum here. So this corresponds to P is equal to 2. So in this way you can understand the meaning of these indexes M, N and P. Now similarly for the TE mode, so the definitions of M and P are almost similar. So here EZ is equal to 0 and you have the BZ field. And in the cavity remember when you have the BZ field, unlike the EZ field the BZ does not go to 0 at R is equal to RC. So here BZ is the tangential component which by boundary condition is allowed. So it will not be 0 here at R is equal to RC. So it will have some finite value at R is equal to RC. So here again M is the number of full period variations in theta of the field components. This is same as before, N is the number of zeros of the actual field components in the radial directions in the range this. So same definition with there are some exceptions which I will explain. P is equal to now P takes value 1, 2, 3 so on it is the number of half period variations in Z of the fields. Now the difference here is from the TM mode that P does not take value 0 which we will see in a moment. So P cannot be equal to 0. So here M is equal to 0, 1, 2 it is the number of full period variations in theta of the field components. So M is equal to 0 means any field component you take no variation in the theta direction. M is equal to 1 means 1 full period variations and so on. N takes value 1, 2, 3 so on it is the number of zeros of the actual field component in the radial direction. So here as I was saying BZ will not be 0 at R is equal to RC. It will have a finite value at R is equal to RC. And again which Bessel function to C depends upon the value of M. So in this case for example we are seeing the value of J0 here J1. P takes value 1, 2 so on it is the number of half period variations in Z of the field. So P is equal to 0 is not possible. In other words a TEMN0 mode cannot exist inside the cavity. So P is equal to 1 means 1 half period variation, P is equal to 2 means 2 half period variations and so on. Okay, so now let us see why P is equal to 0 is not possible. Let us imagine for a moment that P is equal to 0 is possible. So it is a TE mode so you have BZ field okay, BZ is equal to 0 you have a BZ field. So you have a BZ field and P is equal to 0, P is equal to 0 means what? It means that there is no variation in the, in the, in any of the field components or there is no variation in BZ along the Z direction okay. Now at these boundaries, at the end walls it is, this is a conductor at these walls if you see BZ it is the normal component, BZ is the normal component and by boundary condition it is not allowed. So BZ has to go to 0 at Z is equal to 0 and Z is equal to L. So if BZ is to be constant along Z so here it is 0 and here it is 0 and it is to be if P is equal to 0 it has to be constant along Z so that means BZ will be 0. So now for TE mode EZ is already equal to 0 and if P is equal to 0 BZ also goes to 0. So now you know that ERE theta, BRB theta they depend upon the values of EZ and BZ. So all fields will go to 0 in other words there is no mode for P is equal to 0. So hence that is why TE MN0 mode cannot exist in a pillbox cavity or a cylindrical cavity. Now again very quickly let us see the modes in the T some TE modes in the cylindrical cavity. So we have TE 111. So if we see so here N is equal to 1 so that means 1 full period variations in the field in the theta direction. So if you see the electric field here and if I plot ER so this is an ER with theta from 0 to 2 pi okay so ER is 0 at this point okay here it is 0 at this point ER is in one direction and some maximum value again going to 0 here and then again it is maximum here in the opposite direction and then going to 0. So you see that it has one full period variation in going from 0 to 2 pi okay. So this means that M is equal to 1. Now since M is equal to 1 for seeing the value of N we have to see the variation of J1. So if you plot J1 is 0 at r is equal to 0 and it need not be 0 here at r is equal to rc because by boundary condition it is the tangential component of magnetic field and it is allowed. So the variation is like this okay. Now so this is an exception that to the rule there that it is the number of zeros N is the number of zeros from 0 to rc excluding the zero at this because now N here cannot take value 0. N is the it is the nth root of the Bessel function Jm. So it cannot take a value 0. So N is equal to 1 here. Now P is equal to 1 means if you see the magnetic field here in this case in the z direction okay so you will see so you will see here that it is equal to 0 here and it is equal to 0 here. So there is no component of Bz here okay because Bz is not allowed at this boundary at these boundaries Bz is not allowed. So here there is Br component then going to Bz and so if you see the variation of Bz with z from 0 to L you have one half period variation. Similarly Te211 mode so here you have two full period variations in Er in going from 0 to 2 pi. So you can see from here in Er so it is 0 here, 0 here, 0 here, 0 here okay. So you have two full period variations in. So here M is equal to 2 so you have to see the variation of Bessel function J2 and N is equal to 1 here again P is equal to 1 here same as before. So if you see the Te211 mode so no variation in the field in the theta direction and now you have to see the variation of because M is equal to 0 so you have to see the variation of J0 so from 0 to Rc so you see that it is maximum here going to 0 here and then again maximum in the opposite direction so which you can see from here it is maximum at R is equal to 0 going to 0 and then in the opposite direction here and P is equal to 1 so you have one half period variation in the z direction. Then similarly you can also have a Te211 mode so you have three full period variations in the theta direction now you have to see the variation of J3 with R and P is equal to 1 a Te112 mode so again here one full period variations in theta and in the z direction there is two half period variations in the field a Te212 mode so here you have two full period variations you see the variation of J2 and here one two half period variations in the z direction as above so this is how you understand the modes in the cavity so depending on the values of M, N and P different types of modes are possible in the cavity.