 Hello and welcome to lecture 3 of the second module of this course on accelerator physics. So here again we will learn about RF acceleration, but before that let us just revise what we did in the last lecture. So we saw that how with time varying fields a small voltage can be used repeatedly to accelerate to high energies by successively accelerating charge particles over many years. So unlike a DC accelerator we need not generate a very high voltage, we can generate a small voltage and use it several times for accelerating to high energies. The necessary condition is isochronism that means particle arrives at each gap at the right time to see the right phase of the electric field and get accelerated and there should be a component of electric field in the direction of velocity of the beam. So without isochronism we can get acceleration in any of the gaps, but if you want to achieve sustained acceleration over several gaps then isochronism is important. So this is known as the principle of successive acceleration. We do not use the entire positive cycle for acceleration, we use only a small portion because if you use the entire cycle then you will get large deviation in kinetic energy for the particles which we do not hold. So we saw that even though the entire positive cycle produces acceleration in this region that means from 0 to pi by 2 there is no phase stability. So if you want to accelerate with phase stability you have to choose your synchronous phase as lying between minus pi by 2 and 0. So the synchronous phase must be chosen to lie between minus pi by 2 and 0. We also saw that the energy gain in a time varying field is less than the energy gain in a DC field by a factor of T where T is known as the transit time factor. So it is a dimensionless quantity and it comes into picture because the charged particles take a finite time to transit over the gap and during that time the field in the gap is varying with time. So that is why this transit time factor comes into picture and T is always less than 1. So the energy gain in another field is always less than the energy gain in a DC field. We also saw that the first accelerator which was designed by Ising and Widrow it had issues when accelerating to higher energies because there an evacuated glass cylinder was used and inside that there were drift tubes to which voltage was applied directly. So since voltage was applied directly to the drift tubes what happened was that so there the cell length was equal to beta lambda by 2. So now as beta increased in order to keep the cell length reasonable lambda had to be decreased or in other words the frequency had to be increased. And at higher frequencies the system started radiating like an antenna. So when the length of the drift tubes becomes comparable to the wavelength of the applied RF instead of storing energy in the gap the drift tubes start radiating energy. So this structure failed to accelerate charged particles at higher energies. So Widrow to overcome this problem Alvarez came up with a solution he said to use the electric fields associated with the electromagnetic waves ok. Now you know that we saw in the last lecture that electromagnetic waves can be so you can derive it from the Maxwell's equation in free space. So they are TEM type of waves and we have an electric field and magnetic field associated with it. So he said that use the electric fields associated with the electromagnetic wave inside a high Q cavity. So we will see over the next few lectures that how inside a high Q cavity the electric field of the electromagnetic wave can be used for acceleration. So just summarizing again from the last lecture electromagnetic waves in free space are transverse electric magnetic waves. So here the electric field the magnetic field and the propagation constant they are mutually perpendicular to each other ok. So electric field has a sinusoidal variation with time. However they cannot these electromagnetic waves in free space they cannot be used for acceleration because for interaction of the wave with the beam the beam should be propagating in the. So beam is propagating in the direction of the electromagnetic wave and electric field will always be perpendicular to the beam. So that is why E dot V is equal to 0. So that is why we will not get any acceleration using a electromagnetic wave in free space. Ok, now let us see what happens to electromagnetic waves bounded by conducting boundaries ok. So let us say we have this electromagnetic wave which is propagating with a propagation constant k and now on both these sides I put conducting boundaries ok. So this is a conductor. So now what happens now after this the wave can no longer propagate so it gets reflected. So it will travel inside this by multiple reflections. So it will travel like this ok. Now I can have two cases here ok. So in the first case let me call it Tm transverse magnetic. The magnetic field is transverse it is coming out of the screen the magnetic field is transverse to the direction of propagation. The net direction of propagation is now k prime which is in this direction because the wave is now travelling with multiple reflections the net direction will be k prime which is this direction. So let us see the Tm wave. So here the magnetic field is perpendicular to the direction of propagation ok. So since the electric field, magnetic field and k vector are mutually perpendicular so the electric field in this case will be in this direction ok. Now I can resolve this electric field such that I have a component which I call E parallel and E perpendicular. So parallel to the direction of propagation and perpendicular to the direction of propagation. So now I see that when this electromagnetic wave is travelling inside the space bounded by conducting boundaries for the Tm wave I have been able to generate a component of electric field along the direction of propagation ok. So there is a now a component of electric field in the direction of propagation. So if I put in let us say a beam here then at least this condition that the electric field and the should be along the direction of velocity of the beam is satisfied. So I may be able to use this for acceleration. Now we can have another case which is the Te wave. So here E is perpendicular to k prime. So here the again the wave is travelling by multiple reflections inside this medium bounded by the conducting boundaries. Now here the electric field is perpendicular to the direction of propagation it is perpendicular to the direction of propagation it is coming out of the screen. In this case the magnetic field will be in this direction since E, B and k have to be mutually perpendicular so the magnetic field will be like this and I can resolve the magnetic field again into two components B parallel and B perpendicular. So I have a component of magnetic field in the direction of propagation but this will not help me in acceleration here the electric field is still perpendicular to the direction of propagation. So if I put in a charge particle beam here it will always see an electric field which is perpendicular to the velocity of the charge particle. So I will not be able to use this for acceleration ok. So in free space bounded by conducting boundaries electromagnetic waves they now propagate as TEM or TE waves they do not propagate as TEM waves. In free space the electromagnetic wave was propagating as a TEM wave but now when it is in a space bounded by conducting boundaries they are going to be propagating as TEM or TE waves. So in TEM waves there is a component of electric field so there is this component of electric field along the direction of propagation ok. So it may be possible ok let us see whether it is actually possible for now let us say it may be possible to use TEM waves for acceleration. So before proceeding ahead let us understand the difference between the travelling waves and standing waves in a simple one dimension system. So a travelling wave in one direction here it is shown by the green curve here. So it is travelling in the forward direction the expression is written like this. So it is propagating wave in the x direction. So this wave hits the boundary and it is reflected back the reflected wave is the blue wave. So it can be written like this it is still a travelling wave. So here you can see that it is travelling in the x direction. The resultant wave is in the is a superposition of the two waves the two standing waves. So if you take the you simply add up the two here the forward wave and the reflected wave so you will get the standing you will get a standing wave which is the net resultant wave. Here if you simplify this and rearrange we see that the space variation and time variation are separated. So it is a standing wave this wave the red wave is not propagating with time it is only oscillating with time. So there are locations here where the amplitude is 0 and at certain so these are known as nodes and there are locations where the amplitude is maximum and oscillating with time. At other locations the amplitude is in between 0 and the maximum amplitude but the frequency of oscillation of all the points is the same. So this is a standing wave we can find out the location of the nodes and the anti nodes. So for the nodes the motion is 0 so y is equal to 0. So if you put y is equal to 0 here so you get kx is equal to 0 pi 2 pi and so on or in other words x is equal to 0 lambda by 2 lambda 3 lambda by 2 and so on. Similarly so there is no motion of these points similarly we can find out the location of the points with maximum amplitude that is y is equal to 2a. So if you substitute y is equal to 2a so you get kx as equal to pi by 2 3 pi by 2 and so on. So these are at locations x is equal to lambda by 4 x is equal to 3 lambda by 4 and so on. So this corresponds to a travelling wave and this corresponds to a standing wave. Now in order to understand the electromagnetic the behaviour of the electromagnetic waves in a cavity we need to understand the concept of modes. So let us first consider a very simplistic system a mechanical system. So let us consider the waves in a string. So suppose you have a way you have a string and you simply swing it like this so what you get is a travelling wave like this so it has an expression which is like this. Now suppose you apply a boundary condition that the string is fixed at both the ends. Now it is no longer a travelling wave because now you have applied these boundary condition that the displacement at both these ends is 0 so not all waves are allowed it is no longer a travelling wave so only certain waves or wavelengths that just fit in only they are allowed. So you can have a case like this where your L is equal to your lambda by 2 this is lambda by 2 this is known as the fundamental mode you can calculate the frequency of the fundamental mode which is V by 2L. You can have another case where L is equal to lambda still it is satisfying the boundary condition that the displacement at the 2 ends is 0. So you can have a condition where L is equal to lambda so this is known as the second harmonic again you can calculate the frequency it is V by L 2 times the frequency of the fundamental mode. Another condition that you can get is L is equal to 3 lambda by 2 so this is 3 lambda by 2 and this is equal to L so this is the third harmonic okay and again you can calculate the frequency it is 3 times the fundamental frequency and so on. So now when you have applied this boundary condition that the ends of the string are fixed okay they have always to satisfy this boundary condition not all wavelengths are allowed only those wavelengths that satisfy the boundary condition only they are allowed okay. So each mode of vibration here okay this each of these modes of vibration this is called a mode and associated with each mode there is a fixed field pattern so for example for this the field pattern is like this this is varying with time there is time variation but no space variation and so with each mode this is the first mode or the fundamental mode then the second harmonic third harmonic and so on associated with each mode there is a fixed field pattern and a fixed frequency okay. Also if you see the frequencies of the higher modes they are integral multiples of the fundamental mode okay and the fundamental mode frequency if you see it depends upon the dimensions of the system so this length here is L so it depends upon the dimensions of the system. So when you apply boundary conditions to a travelling wave you get what is known as modes. So the first mode is the fundamental mode it has a frequency that depends only on its dimensions okay so you see it is depending only on the dimension the frequencies of the higher order modes they are integral multiples of the fundamental modes so n is an integer here so it is integral multiple of the fundamental mode. Then all points on the string they will oscillate at the same frequency but different amplitude so this point for example it is fixed now this will oscillate at a different amplitude as compared to this point but at the same frequency. So associated with each mode is a fixed field pattern and a fixed frequency so applying boundaries has now made this travelling wave into a standing wave and in the direction that the boundaries have been applied it has become a standing wave and it can take only certain discrete wavelengths. So similarly when you have electromagnetic waves and they are put inside a region bounded by conducting boundaries okay they have to satisfy the boundary conditions at the boundaries and modes are formed. So for the case of electromagnetic waves also it is a similar case that when you so an electromagnetic wave in free space it is a travelling wave when you put it in a region with conducting boundaries at the boundary of the conductors it has to satisfy certain boundary conditions so it satisfies those boundary conditions and in that direction it becomes a standing wave okay so what are the boundary conditions. So for a perfect conductor where the conductivity is infinite so at the boundary of the conductor the fields cannot penetrate inside the conductor okay so at the boundary of the conductor we have the tangential component that means the component which is parallel to the boundary and the normal component that means the component which is perpendicular to the boundary. So the boundary conditions at this interface of conductor and free space are the tangential component at the surface of the conducting plane is 0 that means E tangential is equal to 0 okay E normal is allowed there is no problem with E normal but E tangential is equal to 0 E normal is not equal to 0 okay similarly for magnetic field the normal component of magnetic field at the surface of the conducting plane is 0. So we have B normal is equal to 0 okay now B tangential is not equal to 0 so this is the boundary condition for electric field and magnetic field whenever it comes at the boundary of a conducting surface okay so in order to understand the propagation of electromagnetic waves in conducting boundaries let us first consider a very simple case where we are applying boundaries only in one dimension okay so here we have two infinite parallel conductors okay so these are two infinite parallel conducting planes okay in the y direction so there is one at y is equal to 0 and there is another one at y is equal to a. So these are infinite conducting parallel planes okay it is a hypothetical system but this is just to understand the concept of the propagation of electromagnetic waves so let us say we have an incident wave so the incident wave can be represented by this expression it is a traveling wave so we already derived the expression so we have an incident wave now for simplicity let us take that the incident wave so this is the direction of propagation okay it is in the yz plane so you can see from here it is in the yz plane okay so k is given by k this angle is theta so this is k cos theta and this is k sin theta so you can write this as k can be written as k cos theta y plus k sin theta unit vector z where k is omega by c where omega is the frequency of the wave in free space and c is the velocity of light so this is the expression for the electromagnetic wave and it is traveling the propagation constant the direction is given as this okay so here k dot r r is simply the position vector so you can write it as x unit vector x plus y unit vector y plus z unit vector z so simply multiply k dot r okay and you will get this expression so you can see that it is a traveling wave in the y direction and in the z direction okay so it is a traveling wave in both y direction and z direction so this wave now it will travel by multiple reflections so it will go like this and then it will get reflected it will get reflected so the net direction of propagation will now be along z okay so let this represented wave reflected wave let me represent it by this wave here okay so it will make the same angle theta here with the y axis okay and I can write this for the reflected wave I can write the wave as j k prime dot r minus omega t okay so for k prime the distance of medium is the same so their amplitudes will be the same okay and I can write this as now this will be if this angle is theta this will be minus k cos theta and this will be k sin theta again multiplying by the position vector the reflected wave will have a form like this okay now for this two polarizations of the electromagnetic wave are possible okay so this is the electromagnetic wave propagating in this direction now I can have an electric field so I can have an electric field which is along the x direction okay so for this wave I can have a case where the electric field is along the x direction okay so this is transverse to the direction of propagation so I called it the Te wave okay I can also have I can also have the magnetic field along the x direction in this case the magnetic field is perpendicular to the direction of propagation so this is known as the Tm type of wave so both cases are possible I can have either a Te wave or I can have a Tm wave depending on what is the direction of E or V